TI-Basic, 80 bytes

" 
Repeat 99<length(Ans
Ans+"-
End
Ans→Str1
Repeat 0
getTime
ClrDraw
Text(0,0,Ans(1),sub(Str1,1,1+Ans(2
Text(6,0,Ans(1)+1,sub(Str1,1,61-Ans(2
End

Relatively straightforward. That's a string with one space at the beginning. The hours are right-aligned. This only works on TI-84+ calculators since the TI-83 does not have an internal clock.

Edit: Thanks @kundor for noticing that I didn't close the last loop. Fixed now (+2 bytes).

Batch, 197 bytes

@echo off
set/ah=100+%time:~0,2%,m=1%time:~3,2%
cls
call:l
set/ah=(h-3)%%24+100,m=260-m
call:l
timeout/t>nul 60
%0
:l
set s=%h:~1% 
for /l %%i in (101,1,%m%)do call set s=%%s%%-
echo %s%

Note: 10th line has a trailing space. For me, %time% formats hours with a leading space but minutes with a leading zero. I decided a leading zero was an easier output format, since all I have to do for that is to add 100 hours and remove the first digit. Minutes are trickier as 08 or 09 will cause octal parse errors, so I prefix a 1 effectively adding 100 minutes, adjusting for this by offsetting the loop appropriately, which is a byte shorter than subtracting the 100.

Python 3.6, 110 bytes

from time import*
while[sleep(60)]:h,m=localtime()[3:5];print('\n'*30+'%2d '%h+'-'*m+f'\n{h+1:2} '+'-'*(60-m))

This uses the new f-string formatting to save a few bytes. You can omit ;sleep(60) to save 10 more bytes, but then it just spams the screen.

Saved one byte changing while 1:...;sleep 60 to while[sleep(60)]:..., thanks to TuukkaX.

Perl 6, 113 bytes

loop {$_=DateTime.now;.put for |('' xx 30),|([\+](.hour,1)».fmt('%2s')Z('-' Xx[\-](.minute,60)».abs));sleep 60}

Try it once with a one second timeout.

Or try an altered version that outputs the result of running for several hours.

Expanded:

loop {                  # keep repeating forever

  $_ = DateTime.now;    # assign an object representing the current time

    .put                # print with trailing newline
                        # ( adds a space between numbers and dashes )

  for                   # for each of the following

    |(                  # make it a slip so that it is all part of one list

      '' xx 30          # 30 empty strings (30 empty lines)
    ),

    |(

        [\+](           # triangle produce
          .hour,        # the hour
          1             # the hour plus one

        )».fmt( '%2s' ) # both formatted to two element string ( space padded )

      Z                 # zipped with

        (
            '-'         # a dash

          Xx            # cross (X) using string repeat (x) operator

            [\-](       # triangle produce
              .minute,  # the minute
              60        # the minute minus 60

            )».abs      # absolute value of both
        )
    );

  sleep 60              # wait until the next minute
}

QBasic, 120 127 121 bytes

Don't run this for very long or your laptop will catch fire. Now 99.several9s% more CPU-efficient.

CLS
m=TIMER\60
h=m\60
m=m MOD 60
FOR i=1TO 2
?USING"## ";h MOD 24;
FOR j=1TO m
?"-";
NEXT
?
h=h+1
m=60-m
NEXT
SLEEP 1
RUN

Ungolfed and explanation

DO
    CLS
    totalMinutes = TIMER \ 60
    hour = totalMinutes \ 60
    minute = totalMinutes MOD 60

    FOR row = 1 TO 2
        PRINT USING "## "; hour MOD 24;
        FOR j = 1 TO minute
            PRINT "-";
        NEXT j
        PRINT

        hour = hour + 1
        minute = 60 - minute
    NEXT row

    SLEEP 1
LOOP

We start by clearing the screen, then get the current hours and minutes from TIMER, which returns the number of seconds since midnight.

This is the first time I've tried PRINT USING, so I was delighted to discover that it doesn't suffer from the usual QBasic quirk that positive numbers are printed with a leading space. ## as the format specifier ensures that single-digit numbers are right-aligned and padded with a space, as required. We have to use a loop for the hyphens, unfortunately, since QBasic does not have a string repetition function. (If I'm mistaken, please let me know!)

All the PRINT statements end with ; to suppress the newline; but after the hyphens, we need a newline; thus, the solitary ? after the inner FOR loop.

The SLEEP 1 is now necessary. Without it, the screen gets cleared so quickly after printing that it's just a flickering mess. (I used LOCATE 1 instead of CLS at first for that reason, until I realized that CLS with SLEEP is shorter anyway.) RUN restarts the program from the top--the shortest way to get an infinite loop.

Ruby, 98 95 91 bytes

Updates every 5 seconds. Only works in Unix-style terminals.

loop{t=Time.now;puts`clear`+"%02d %s
%02d "%[h=t.hour,?-*m=t.min,-~h%24]+?-*(60-m);sleep 5}

Windows command prompt version, 95 92 bytes:

loop{t=Time.now;puts"\e[H\e[2J%02d %s
%02d "%[h=t.hour,?-*m=t.min,-~h%24]+?-*(60-m);sleep 5}

Java 8, 313 300 299 bytes

import java.time.*;()->{for(int c=0,h=LocalDateTime.now().getHour(),m=LocalDateTime.now().getMinute(),i;;)if(c>30){c=0;String l="",u,d;for(i=0;i++<60;)l+="-";u=l.substring(0,m);d=l.substring(m);System.out.println((h<10?"0":"")+h+" "+u+"\n"+(h<9?"0":"")+(h+1)+" "+d);}else{c++;System.out.println();}}

This only updates every 30 iterations of the while loop. The other 29 iterations just print new lines.

Updated

Saved 13 14 bytes due to Kevin Cruijssen's help! Thanks!

Python 2, 131 129 127 bytes

from time import*
while[sleep(9)]:exec(strftime("a='%H';b=int('%M')"));print "\n"*30+a+" "+"-"*b+"\n"+`int(a)+1`+" "+"-"*(60-b)

saved a byte thanks to @TuukkaX

GameMaker Language, 134 bytes

s=" "while 1{s+="-"a=current_hour b=current_minute draw_text(0,0,string(a)+string_copy(s,1,b+1)+"#"+string(a+1)+string_copy(s,0,61-b)}

In the settings, you must be ignoring non-fatal errors in order for this to work. Also, in GML, # is equivalent to \n in most languages.

Perl 6, 104 bytes

DateTime.now.&{"\ec{.hour.fmt: '%2s'} {'-'x.minute}\n{(.hour+1).fmt: '%2s'} {'-'x 60-.minute}"}.say xx*

Needs to be run on a ANSI compatible terminal so that the control sequence for resetting the terminal works.

Pretty basic (because the more obfuscated approaches I tried turned out longer):

• DateTime.now.&{" "}.say xx*: Transform the current time into a string (see below) and say it, and repeat all of that an infinite number of times. The string is built like this:
  • \ec: ANSI control code <ESC>c for resetting the terminal, which clears the screen.
  • {.hour.fmt: '%2s'}: hour, right-aligned to 2 columns
  • : space
  • {'-'x.minute}: dash repeated times the minute
  • \n: newline
  • {(.hour+1).fmt: '%2s'}: next hour, right-aligned to 2 columns
  • : space
  • {'-'x 60-.minute}: dash repeated times 60 minus the minute

PHP, 112 bytes

for(;;sleep(9))echo($s=str_pad)($h=date(H),99,"\n",0).$s(" ",1+$m=date(i),"-")."\n".(++$h%24).$s(" ",61-$m,"-");

As there's no way to clear the screen (that I can find) I had to go with a pile of newlines. Also the question being updated to "at least" once a minute saves a byte with 9 instead of 60.

JavaScript (ES6), 162 bytes

Updates once per second

setInterval(c=>{c.clear(d=new Date,m=d.getMinutes(),h=d.getHours(),H=_=>`0${h++}`.slice(-2)),c.log(H()+` ${'-'.repeat(m)}
${H()} `+'-'.repeat(60-m))},1e3,console)

MATL, 38 bytes

`XxFT"4&Z'@+OH&YAO'-'60@*5&Z'-|Y"hhD]T

Try it at MATL online! But note that the program is killed after 30 seconds, so it's difficult to catch any changes in the output.

How it works

`           % Do...while
  Xx        %   Clear screen
  FT        %   Push [0 1]
  "         %   For each k in [0 1]
    4&Z'    %     Push current hour
    @+      %     Add k
    OH&YA   %     Convert to base-10 string with 2 digits
    O       %     Push 0. Concatenating with chars will convert this to char,
            %     and char(0) will be displayed as a space
    '-'     %     Push '-'
    60@*    %     Push 60*k
    5&Z'    %     Push current minute
    -|      %     Absolute difference. This gives current minute for k==0,
            %     or 60 minus that for k==1
    Y"      %     Repeat '-' that many times
    hh      %     Concatenate the top three elements into a string
    D      %      Display
  ]         %   End
  T         %   Push true
            % End (implicit). Since the top of the stack contains true, this
            % gives an infinite loop

PHP, 104 bytes

<? for(;;sleep(6))printf("%'
99s%2d %'-".($m=date(i))."s
%2d %'-".(60-$m)."s","",$h=date(H),"",++$h,"");

showcase for printf´s custom padding characters:
"%'-Ns"=left pad string with - to N characters.

will print 99 newlines (every 6 seconds) instead of clearing the screen.

First newline must be a single character. So, on Windows, it must be replaced with \n.

AWK, 190 bytes

#!/bin/awk -f
func p(x,y,c){printf("%2s ",x)
for(j=0;j<y;j++)printf(c)
print}BEGIN{for(;;){split(strftime("%H %M"),t)
m=t[2]
if(o!=m){p(a,30,"\n")
p(t[1],m,"-")
p((t[1]+1)%24,60-m,"-")}o=m}}

Since AWK doesn't have a built-in sleep function, I simply have it continually check the clock to see if the minute has changed yet. The key thing is that it works... right? :)

Python 3.5, 127 120 117 bytes

from time import*
while[sleep(9)]:h,m=localtime()[3:5];print('\n'*88,*['%2d '%x+'-'*y+'\n'for x,y in[(h,m),(h+1,60-m)]])

C 251 267 bytes

 #include<time.h>f(){time_t t;struct tm *t1;h,m,i;while(1){time(&t);t1=localtime(&t);int h=t1->tm_hour;int m=t1->tm_min;printf("%d ",h);for(int i=1;i<=m;i++)printf("-");printf("\n");printf("%d ",h+1);for(int i=0;i<=59-m;i++)printf("-");printf("\n");sleep(1);system("clear");}}

Ungolfed version

#include<time.h>
void f()
{
 time_t t;
 struct tm *t1;
 int h,m,i;

 while(1)
 {
  time(&t);     
  t1=localtime(&t);
  h=t1->tm_hour;
  m=t1->tm_min;

  printf("%d ",h);
  for(int i=1;i<=m;i++)
   printf("-");

  printf("\n");
  printf("%d ",h+1);

  for(int i=0;i<=59-m;i++)
   printf("-");

  printf("\n");    

  sleep(1);
  system("clear");    
 }
}

Gets the work done! Can definitely be shortened in some way. Assume unistd.h file is included.

@Neil Thanks for the info.

Python, 115 113 bytes

_{saved a couple of bytes thanks to @kundor and @Phlarx}

import time
while 1:h,m=time.localtime()[3:5];print("\x1b[0;H{:02} {}\n{:02} {} ".format(h,"-"*m,h+1,"-"*(60-m)))

BASH, 165 141 156 bytes

while :
do
clear
m=`date +%-M`
a=`printf %${m}s`
b=`printf %$((60-$m))s`
h=`date +%H`
echo $h ${a// /-}
printf "%02d %s" $((10#$h+1)) ${b// /-}
sleep 9
done

Gura, 138 bytes

k(a,b)={if(a<10){a="0"+a;}println(a," ","-"*b)};repeat{t=datetime.now();k(t.hour,t.min);k(t.hour+1,60-t.min);os.sleep(60);print("\n"*30);}

Pretty short and straightforward :)

Processing, 204 200 198 bytes

4 bytes saved thanks to @L. Serné by using smarter ternaries

void draw(){int i;String s=((i=hour())>9?i:" "+i)+" ";for(i=0;i<minute();i++)s+="-";s+="\n"+(((i=hour()+1)>24?i=0:i)>9?i:" "+i)+" ";for(i=0;i<60-minute();i++)s+="-";print(s);for(;i++<99;)println();}

This outputs 30+ newlines for each update (which takes place when the frame gets updated)

Golfing in progress...

Ok, haven't done a code golf in a while, so here goes my sad attempt :)

Unix Korn Shell: 177 171 170 bytes

while :
do
clear
h=`date +%H`
m=`date +%M`
d=-----
d=$d$d$d$d
d=$d$d$d
a=`echo $d|cut -b-$m`
let m=60-$m
b=`echo $d|cut -b-$m`
let i=h+1
echo "$h $a\n$i $b"
sleep 9
done

Mathematica, 235 bytes

d=UpdateInterval;e=Dynamic;f=Refresh;g=AbsoluteTime;Grid[Partition[Riffle[e[f[Floor@Mod[g[]/3600+#,24],d->1]]&/@{0,1},With[{t=#},e[f[""<>Array["-"&,If[t==60,60-#,#]]&@Setting@Floor@Mod[g[]/60+#,60],d->1]]]&/@{0,60}],2],Alignment->Left]

C#, 192 185 176 172 bytes

using System;()=>{while(1>0){Console.Clear();var t=DateTime.Now;int h=t.Hour,m=t.Minute;Console.Write($"{h:00} {new string('-',m)}\n{++h%24:00} {new string('-',60-m)}");}};

Uses C# 6 features which I don't have access to so cannot test it properly yet.

Full version showing how it works and is called including a Thread.Sleep so the input can actually be seen. Note that this version shows the use of Console.Writes overload to use String.Format and doesn't use string interpolation like the golfed version above.

using System;

class P
{
    static void Main()
    {
        Action a = () =>
        {
            while (1 > 0)
            {
                Console.Clear();

                var t = DateTime.Now;
                int h = t.Hour,m=t.Minute;

                Console.Write("{0:00} {1}\n{2:00} {3}", h, new string('-', m), ++h % 24, new string('-', 60 - m));

                System.Threading.Thread.Sleep(500);
            }
        };

        a();
    }
}

Python3.5 104 bytes.

from time import*
while[sleep(9)]:h,m=localtime()[3:5];print("{:02d} {}".format(h,m*'-'),end='\n'*30)

Riffing off Gurupad Mamadapur's answer above.

edit: xnor is right. This isn't a valid answer. - This misses the second line completely.

C# Interactive (138 Bytes)

while(true){var d=DateTime.Now;Console.WriteLine($"{d.Hour:00} {new string('-',d.Minute)}\n{d.Hour+1:00} {new string('-',60-d.Minute)}");}