Python 2, 43 45 bytes

lambda s:reduce(lambda p,n:n*(7-p!=n!=p>0),s)

43 bytes (inspired heavily by @Zgarb)

lambda s:reduce(lambda p,n:n*(p>0<n^p<7),s)

This function combines my reduce statement with the bit-flicking logic from @Zgarb's answer, for a combination that is shorter than both.

Both answers output the following:

• 0 if the input is not a valid sequence
• The last digit of the sequence if it is valid

Python, 44 bytes

lambda x:all(0<a^b<7for a,b in zip(x,x[1:]))

Bitwise magic! This is an anonymous function that takes a list of integers, and checks that the XOR of every two consecutive elements is between 1 and 6 inclusive.

Why it works

First, the XOR is always between 0 and 7 inclusive, since 7 is 111 in base 2, and our numbers have at most 3 binary digits. For the equality, a^b == 0 if and only if a == b. Also, we have 7-a == 7^a when 0 ≤ a ≤ 7, and thus a^b == 7 if and only if a == 7^b == 7-b.

05AB1E, 11 9 bytes

-2 bytes for Osable's smart idea of using a product.

¥¹D7-Á+«P

Try it online!

¥           # Push deltas.
 ¹D7-Á      # Push original array, and 7 - [Array] shifted right once.
      +     # Add original to the 7 - [Array] shifted right.
       «    # Concat both.
        P   # Product, if either contain a zero, results in 0, meaning false.

Third approach using 05AB1E, that doesn't use the pairwise command:

• 0 if it violates the tipsy properties.
• Not 0 if there was nothing preventing it from being tipsy.

05AB1E, 10 bytes

$ü+7ʹüÊ*P

Uses the CP-1252 encoding. Try it online!

R, 39 37 32 bytes

q=diff(x<-scan())
all(q,2*x+q-7)

Takes input from stdin. Uses diff to see if any two consecutive digits are the same; then compares each digit to 7 minus the previous digit. Returns TRUE or FALSE.

Saved 5 bytes thanks to Jarko Dubbeldam.

05AB1E, 15 bytes

ü+7å¹ü-0å¹g_~~_

Try it online! or as a Test suite

JavaScript (ES6), 43 40 bytes

Returns 0 / true.

f=([k,...a],n=0)=>!k||k-n&&7-k-n&&f(a,k)

Test cases

f=([k,...a],n=0)=>!k||k-n&&7-k-n&&f(a,k)

console.log('Testing truthy cases ...');
console.log(f('1353531414'));
console.log(f('3132124215'));
console.log(f('4142124136'));
console.log(f('46'));
console.log(f('4264626313135414154'));
console.log(f('6'));
console.log(f('2642156451212623232354621262412315654626212421451351563264123656353126413154124151545145146535351323'));
console.log(f('5414142'));

console.log('Testing falsy cases ...');
console.log(f('11'));
console.log(f('3132124225'));
console.log(f('6423126354214136312144245354241324231415135454535141512135141323542451231236354513265426114231536245'));
console.log(f('553141454631'));
console.log(f('14265411'));
console.log(f('16'));
console.log(f('42123523545426464236231321'));
console.log(f('61362462636351'));
console.log(f('62362462636361'));

Perl 6, 22 bytes

Using a regex:

{!/(.)<{"$0|"~7-$0}>/}

Takes the input as a string. Inspired by G B's Ruby answer.
How it works:

• / /: A regex.
• (.): Match any character, and capture it as $0.
• <{ }>: Dynamically generate a sub-regex to be matched at that position.
• "$0|" ~ (7 - $0): The sub-regex we generate is one that matches only the previous digit, or 7 minus the previous digit (e.g. 5|2).
  Thus the overall regex will match iff it finds an invalid consecutive pair of digits anywhere.
• {! }: Coerce to a boolean (causing the regex to be matched against $_), negate it, and turn the whole thing into a lambda (with implicit parameter $_).

Perl 6, 38 bytes

Using list processing:

{all ([!=] 7-.[1],|$_ for .[1..*]Z$_)}

Takes the input as an array of integers.
How it works:

• .[1..*] Z $_: Zip the input list with an offset-by-one version of itself, to generate a list of 2-tuples of consecutive digits.
• [!=] 7 - .[1], |$_: For each of those, check if (7 - b) != a != b.
• all ( ): Return a truthy or falsy value depending on whether all loop iterations returned True.

JavaScript 61 43 bytes

Comments have mentioned I can't use C# linq functions without including the using statement, so here's the exact same in less bytes using standard JS...

f=a=>a.reduce((i,j)=>i>6|i==j|i+j==7?9:j)<7

C#, 99 67 65 bytes

Takes input as an int array a

// new solution using linq
bool A(int[]a){return a.Aggregate((i,j)=>i>6|i==j|i+j==7?9:j)<7;}

// old solution using for loop
bool A(int[]a){for(int i=1;i<a.Length;)if(a[i]==a[i-1]|a[i-1]+a[i++]==7)return false;return true;}

Explanation:

// method that returns a boolean taking an integer array as a parameter
bool A(int[] a) 
{
    // aggregate loops over a collection, 
    // returning the output of the lambda 
    // as the first argument of the next iteration
    return a.Aggregate((i, j) => i > 6 // if the first arg (i) > than 6
    | i == j      // or i and j match
    | i + j == 7  // or i + j = 7
    ? 9   // return 9 as the output (and therefore the next i value)
    : j   // otherwise return j as the output (and therefore the next i value)
    ) 
    // if the output is ever set to 9 then it will be carried through to the end
    < 7; // return the output is less than 7 (not 9)
}

R, 49 44 bytes

!any(rle(x<-scan())$l-1,ts(x)==7-lag(ts(x)))

Reads input from stdin (separated by space) and outputs TRUE/FALSE. Will give a warning if input is of length one but still works.

Edit: saved a couple of bytes thanks to @rturnbull

Ruby, 34 bytes

->s{!s[/[16]{2}|[25]{2}|[34]{2}/]}

><> (Fish) 47 bytes

0:i:1+?!v\!
   0n;n1< >
!?-{:-"0"/^
!? -{-$7:/^

Pretty simple;

Line 1: check to see if inputted a number, if no number (EOF) then we have a truthy to print else checks.

Line 2: print outcome.

Line 3: turn input into number (ASCII 0 - from input), then check if it's equal to previous input.

Line 4: check if input is opposite side of die.

Brain-Flak 128 Bytes

(()){{}(({}<>)<>[({})])}{}([]){{{}}<>}{}([]){{}({}({})<>)<>([][()])}{}(<{}>)<>(([])){{}{}({}[(()()()){}()]){<>}<>([][()])}({}{})

Outputs 0 for falsey, or -7 for truthy.

Try it Online! (Truthy)
Try it Online! (Flasey)

Explanation (t stands for top and s stands for second from the top):

(())                # push a 1 to get this loop started
{{}                 # loop through all pairs, or until 2 are equal
(({}<>)<>[({})])    # pop t, push t on the other stack, and t - s on this one
}{}                 # end loop and pop one more time
([])                # push the height of the stack
{                   # if the height isn't 0 (there were equal numbers)...
{{}}<>              # pop everything from this stack and switch
}                   # end if
{{}                 # for every pair on the stack: pop the height and...
({}({})<>)<>        # push t + s on the other stack leaving s on this one
([][()])            # push the height - 1
}                   # end loop when there is only 1 number left
{}(<{}>)<>          # pop t, pop s, push 0 and switch stacks
(([]))              # push the height twice
{                   # loop through every pair
{}{}                # pop the height and what was t - 7
({}[(()()()){}()])  # push t - 7
{<>}<>              # if t is not 0 switch stacks and come come back
                    # if t is 0 (ie, there was a pair that added to 7) just switch once
([][()])            # push height - 1
}                   # end loop
({}{})              # push t + s (either 0 + 0 or 0 + -7)

MATLAB, 30 bytes

@(a)all(diff(a)&movsum(a,2)-7)

PowerShell, 57 44 41 bytes

_{(Crossed out 44 is still regular 44)}

0-notin($args|%{7-$_-$l-and$l-ne($l=$_)})

Try it online!

(OP has clarified that taking input as separate arguments is OK - saved 13 bytes ... saved another 3 bytes by eliminating $b)

We're looping through the input $args a digit at a time. Each digit, we verify that the $last digit is -notequal to the current digit $_, and that 7-$_-$l is some number other than zero (which is truthy). Those Boolean results are encapsulated in parens and fed into the right-hand operand of the -notin operator, checking against 0. In other words, if there is any False value anywhere in the loop, the -notin will also be False. That Boolean is left on the pipeline, and output is implicit.

Lengthy because of the $ requirement for variable names, and that Boolean commands -ne -and are verbose in PowerShell. Oh well.

Python, 38 bytes

f=lambda h,*t:t==()or 7>h^t[0]>0<f(*t)

A recursive function that takes arguments like f(1,2,3).

This makes use of argument unpacking to extract the first number into h and the rest into the tuple t. If t is empty, output True. Otherwise, use Zgarb's bit trick to check that the first two die rolls are not incompatible. Then, check that the result also holds on the recursive call on the tail.

C 47 bytes

F(char*s){return!s[1]||!((*s^s[1])%7)&&F(s+1);}

takes a string of digits (or a zero terminated array of bytes)

Explanation

F(char*s){

according to the standard int return type is implied. (saving 4 bytes)

return unconditional return because this is a recursive function

using shortcut evaluation:

!s[1]|| if the second character is nul return true

((!(*s^s[1])%7)&& if the first two charcters aren't legal false

F(s+1)) check the rest of the string in the same way

that confusing expression

*s is the first character s[1] is the second

*s^s[1] exclusive-ors them together if they are the same the result is 0 if they add to 7 the result is 7 ,

so (*s^s[1])%7 is zero for bad input and non-zero otherwise

thus !((*s^s[1])%7) is false if these 2 characters are bad

Processing, 93 92 90 bytes

Changed || to | : 1 byte saved thanks to @ClaytonRamsey

Started counting backwards: 2 bytes saved thanks to @IsmaelMiguel

int b(int[]s){for(int i=s.length;--i>0;)if(s[i-1]==s[i]|s[i-1]==7-s[i])return 0;return 1;}

Takes input as an array of ints, output 1 for true or 0 for false.

Ungolfed

int Q104044(int[]s){
  for(int i=s.length;--i>0;)
    if(s[i-1]==s[i]|s[i-1]==7-s[i])
      return 0;
  return 1;
}

Python, 71 bytes

f=lambda s:len(s)<2or(s[0]!=s[1]and int(s[0])!=7-int(s[1]))and f(s[1:])

Uses a recursive approach.

Explanation:

f=lambda s:                                                              # Define a function which takes an argument, s
           len(s)<2 or                                                   # Return True if s is just one character
                      (s[0]!=s[1]                                        # If the first two characters matches or...
                                 and int(s[0])!=7-int(s[1])              # the first character is 7 - the next character, then return False
                                                           )and f(s[1:]) # Else, recurse with s without the first character

Retina, 28 bytes

M`(.)\1|16|25|34|43|52|61
^0

Try it online!

Alternatively:

M`[16]{2}|[25]{2}|[34]{2}
^0

Try it online!

MATL, 9 bytes

dG2YCs7-h

The input is an array of numbers representing the digits.

The output is a non-empty array, which is truthy if all its entries are non-zero, and falsy otherwise (read more about MATL's criterion for truthy and falsy here).

Try it online! Or verify all test cases.

Explanation

d     % Take input implicitly. Consecutive differences
G     % Push input again
2YC   % Overlapping blocks of length 2, arranged as columns of a matrix
s     % Sum of each column
7-    % Subtract 7, element-wise
h     % Concatenate horizontally. Implicitly display

PHP, 63 bytes

for($d=$argv[$i=1];$c=$argv[++$i];$d=$c)$d-$c&&$d+$c-7?:die(1);

takes input as list of command arguments; exits with 1 (error) if input is invalid, 0 (ok) if valid.

Run with -nr.

input as string argument, 65 bytes

for($d=($s=$argv[1])[0];$c=$s[++$i];$d=$c)$d-$c&&$d+$c-7?:die(1);

C# (with Linq) 90 81 73 71 69 68 Bytes

using System.Linq;n=>n.Aggregate((p,c)=>p<9|c==p|c==103-p?'\b':c)>9;

Explanation:

using System.Linq;           //Obligatory import
n=>n.Aggregate((p,c)=>       //p serves as both previous character in chain and error flag
    p<9                      //8 is the error flag, if true input is already invalid            
        |c==p            
            |c==103-p        //103 comes from 55(7) + 48(0)
                ?'\b'       //'\b' because it has a single digit code (8)
                    :c)      //valid so set previous character, this catches the first digit case as well
                        >8;  //as long as the output char is not a backspace input is valid

C, 81 bytes, was 85 bytes

int F(int *A,int L){int s=1;while(--L)s&=A[L]!=A[L-1]&A[L]!=(7-A[L-1]);return s;}

Input is an array of integers A with length L. Returns 1 for true and 0 for false. The input is checked from the end to the start using the input length L as the array index.

Groovy, 61 bytes

{i=0;!(true in it.tail().collect{x->x in [7-it[i],it[i++]]})}

Python 2, 58 Bytes

lambda x:all(x[i]!=x[i+1]!=7-x[i]for i in range(len(x)-1))

><>, 39 bytes

0>i'0'-::&0)?v1n;>&!
?=0+=7+@=}:{:<;n0^

Try it online!

Batch, 102 bytes

@set s=%1
@set/an=0%s:~0,2%,r=n%%9*(n%%7)
@if %r%==0 exit/b
@if %n% gtr 6 %0 %s:~1%
@echo 1

Ungolfed:

@echo off
rem grab the input string
set s=%1
:loop
rem convert the first two digits as octal
set /a n = 0%s:~0,2%
rem check for divisibility by 9 (011...066)
set /a r = n %% 9
rem exit with no (falsy) output if no remainder
if %r% == 0 exit/b
rem check for divisibility by 7 (016...061)
set /a r = n %% 7
rem exit with no (falsy) output if no remainder
if %r% == 0 exit/b
rem remove first digit
set s=%s:~1%
rem loop back if there were at least two digits
if %n% gtr 6 goto loop
rem truthy output
echo 1

Clojure, 55 49 46 bytes

edit 1: using (mapcat ...) instead of (flatten(map ...))

edit 2: using (juxt - +) instead of (juxt = +), checking for values 0 or 7

#(not(some #{0 7}(mapcat(juxt - +)(rest %)%)))

Consecutive values are not allowed to be equal (difference = 0) or sum up to 7. Got to use juxt again :)

JavaScript (ES6), 46 Bytes

a=>r=1&&a.reduce((c,p)=>r=c==p||c==7-p?0:1)&&r

f=a=>r=1&&a.reduce((c,p)=>r=c==p||c==7-p?0:1)&&r;

console.log(f([4,6]));
console.log(f([1,6]));