Python 2, 126 bytes

Try it online!

import random,string
def f(a,b):
 while b/a:print[a,eval('random.choice(string.printable[10:-6])+'*len(`a`)+"''")][a%8<1];a+=1

Still golfing down.

Many thanks to Flp.Tkc and EasterlyIrk for all of their help!

Mathematica, 96 bytes

Range@##/.a_?(8∣#&):>Join[33~(c=CharacterRange)~47,58~c~127]~RandomChoice~⌊Log10@a+1⌋<>""&

Explanation

For inputs m and n:

Range@##

Generate {m, m + 1, m + 2, ... , n}

/.a_?(8∣#&):>

For all numbers that are divisible by 8 (call that a), apply this replacement rule:

Join[33~(c=CharacterRange)~47,58~c~127]

Get a list of all printable ASCII characters, except digits.

... ~RandomChoice~⌊Log10@a+1⌋

Pseudo-randomly choose Floor[Log10[a] + 1] characters from the list, allowing duplicates.

<>""

Join the characters.

zsh, 100 98 bytes

for i in {$1..$2};{((i%8))&&<<<$i||<<<`yes 'shuf -e {!..~}|grep "[^0-9]"|head -c1'|head -$#i|zsh`}

The two input arguments are passed as command line arguments, and the numbers are output on separate lines.

for i in {$1..$2};{   # loop through the range
((i%8))&&             # if the number is not divisible by 8 (i % 8 != 0),
<<<$i||               # output it
<<<`                  # otherwise, output the following:
yes '                 # using `yes' as a golfy loop
shuf -e {\!..\~}      # shuffle the range of printable ASCII (minus space)
|grep "[^0-9]"        # get rid of numbers
|head -c1'            # take the first character
|head -$#i            # obtain a string with that code repeated len(i) times... 
|zsh                  # ... and eval it
`}

Python 2, 121 120 bytes

import random,string
def f(a,b):print(a,`random.sample(string.printable[10:-6]*a,len(`a`))`[2::5])[a%8<1];f(a+1/(a<b),b)

Terminates with an error.

Try it online!

Python 2, 126 bytes

(one does not simply outgolf Dennis)

Seeing as I did a lot of work on heather's answer, I thought I'd post my own solutions too.

import random,string
def f(a,b):
 while b/a:print[a,eval('random.choice(string.printable[10:-6])+'*len(`a`)+"''")][a%8<1];a+=1

This is a function which takes two arguments and prints directly to STDOUT.

127 bytes

import random,string
lambda a,b:[[x,eval('random.choice(string.printable[10:-6])+'*len(`x`)+`''`)][x%8<1]for x in range(a,b+1)]

This is an unnamed anonymous function - to use, assign to a variable (such as f), and then call with f(a, b). This returns the output as a list.

MATL, 26 bytes

&:"@8\?@}6Y24Y2X-Xz@VnT&Zr

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Explanation

&:        % Input a and b (implicit). Push range [a a+1 ... b]
"         % For each k in that range
  @       %   Push k
  8\      %   Modulo 8
  ?       %   If non-zero
    @     %     Push k
  }       %   Else
    6Y2   %     Push string of all printable ASCII chars
    4Y2   %     Push string '0123456789'
    X-    %     Set difference
    Xz    %     Remove space. Gives string of possible random chars
    @Vn   %     Push number of digits of k
    T&Zr  %     Random sample with replacement of that many chars from the string
          % End if, end for each, display (implicit)

Pip, 28 bytes

Fia,b+1Pi%8?i{RC@>PA@`\D`}Mi

Takes the numbers as command-line arguments and prints a newline-separated list of results. Try it online!

Explanation:

                              a,b are cmdline args; PA is string of all printable ASCII
Fia,b+1                       For i in range(a, b+1):
       P                       Print this:
        i%8?i                  If i%8 is truthy (nonzero), i; otherwise:
             {           }Mi   Map this function to the digits of i:
                @>PA           All but the first character of PA (removes space)
                    @`\D`      Find all regex matches of \D (nondigits)
              RC               Random choice from that list of characters
                               The map operation returns a list, which is concatenated
                               before printing

R, 73 bytes

i=scan();x=i[1]:i[2];x[!x%%8]=sample(sapply(c(32:46,58:126),intToUtf8));x

Reads input from stdin and replaces replaces numbers divisible by 8 with a uniformly chosen sample of ascii characters in the range 32...47, 58...126. To draw the random sample we need a vector of characters, unfortunately intToUtf8() returns one string rather than a vector so we also have to vectorize it over the range using sapply.

Pyth, 24 bytes

jm?%d8dsmO-r\~\ jkUT`d}F

Try it online!

Explanation:

jm?%d8dsmO-r\~\ jkUT`d}FQ  # Auto-fill variables
                      }FQ  # Splat inclusive range on the input
 m?%d8d                    # Map over each number, if it isn't divisible by 8 return it
       smO          `d     # for each other number, select a character at random for
                             each of it's digits and then flatten into one string
           r\~\            # Printable ASCII excluding space
          -     jkUT       # Setwise difference with numeric values (remove numbers)
j                          # Join with newlines

Bash + jot, 64 bytes

Golfed

seq $1 $2|sed "$[9-$1%8]~8s/.*/a=&;jot -r -s '' -c \${#a} ! ~/e"

Test

>crazy8 115 123
115
116
117
118
119
&Wn
121
122
123

>crazy8 1 16
1
2
3
4
5
6
7
Y
9
10
11
12
13
14
15
ZR

Pyke, 22 bytes

h1:Fi8%!I`lV~KT>6_<Hs0

Try it here!

Takes input in the form: higher, lower

JavaScript (ES6), 100 bytes

f=(x,y)=>(x+"").replace(/./g,d=>x%8?d:String.fromCharCode(Math.random()*94+33))+(x<y?[,f(x+1,y)]:"")

O.textContent = f(1,200)

<pre id=O>

Those darn built-ins with 23-byte names....

Perl 6, 60 bytes

{map {$_%8??$_!!S:g/./{grep(/\D/,"!".."~").pick}/},$^a..$^b}

Explanation:

• { map { }, $^a .. $^b }: A lambda that takes two arguments, generates the list of integers in that range, and returns it with the following transformation applied to each element:
• $_ % 8 ?? $_ !!: If the element is not divisible by 8, pass it on unchanged. Otherwise...
• S:g/./{ }/: ...replace each character of its string representation with the value generated by this expression:
• grep(/\D/, "!" .. "~").pick: Generate the range of characters between ! and ~ (in Unicode order), filter out digits, and randomly pick one of the remaining characters.

Python 2, 111 125 Bytes

from random import*
def f(a,b):
 for i in range(a,b+1):
  if i%8<1:i=''.join([chr(randint(58,126))for _ in str(i)])
  print i

EDIT:

Thanks @Flp.Tkc for realising I hadn't read the task properly.

Thanks @Caleb for pointing out I could use a few to reduce the byte count.

Thanks @Dennis for pointing out about the fact that numbers can't be included.