If you are having problems with code that does substitution, then you need to actually include that code.

I've always done this type of replacement using something like this:

my %subs = (
    word1   => 'replace word1 with this text',
    word2   => 'replace word2 with this test',
);

while($document =~ s/\[(\w+)\]) {
    my $word = $1;
    my $rep = $subs{$word} || "!Unknown token: $word!";
    $document =~ s/\[$word\]/$subs{$word}/smg;
}
[download]

#!/usr/bin/perl -w 

%hash = ();

open(FILE, "story.txt") || die ;       #opens file story.txt 
                    
while ( $story[$i++] = <FILE> ){ }     #moves story.txt to @story 

close (FILE);                  #closes Story.txt 

foreach $list ( @story ) 
{        
   @temp = split(/ /, $list);    #puts word by word in @temp
   $i=0;
   while ($i <= $#temp)
   {                 
      $word=$temp[$i];                       
      if ( $word =~ /^\[/ && $word =~ /$\]/)      
      {           
         if (! exists $hash { $word } )
            {    
            $auxiliary_word = $word; 
            $auxiliary_word =~ tr/\[/ /; #removes front bracket 
            $auxiliary_word =~ tr/\].,?/ /; #removes rear bracket & pu
+nctuation             
        $capitol = 0;

        if ($word =~ /^\[[A-Z]/)    #checks for capitol letter
        {                
            $capitol = 1;                             
        }
        Replace($word,$capitol);      #function call to enter own word
+s                         
            }
      }
      $i++;
   }
}
Print();                    #print final madlib story

#*****************************************************************
sub Replace                        #user enters own words
{
    print" Please enter a $auxiliary_word: ";
        $new_word = <STDIN>;
    
        if ($capitol == 1 ) 
    { 
        $new_word =~ s/(^[a-z]+)/\u$1/;    #makes first letter capitol
        }    
        chomp ( $new_word );
        $hash { $word } = $new_word;            #adds capitol letter t
+o hash
}     
          
#*********************************************************************
sub Print                        #function to print new story
{  
    foreach $list ( @story )
    {
       @temp = split(/ /, $list);
       $i=0;
       while ($i <= $#temp)
       {    
              
          $word = $temp[$i];    
        if ($word =~ /^\[/ && $word =~ /$\]/)    
              #if ( $word =~ m/\B[[A-Za-z]]/g)    
              {                       
            $temp[$i] = $hash { $word };               
              }
          $i++;        
         }   
          $final = join(' ', @temp);            #joins story together
       print $final;                #prints new story
    }
}
[download]

Strangeness in Cars 

I once owned a [noun] that was [adjective] only when ,
in a [Make-of-car].  Some [plural-noun] thought,
[adverb], that this was [another-adjective], but what did
they [verb].  Anyway, now I am [comparative],
and don't [verb] so much, even when in [make-of-car]
cars.  Do you think that's [another-adjective], or
should I [strange-action-verb]?
[download]

You can reduce the code quite a bit by playing to Perl's strengths. The quick way to load the story into an array (once you've opened the file) is

  my @story = <FILE>;
[download]

You can reduce the amount of work you're doing to pull out bracketed words by doing

  foreach my $line ( @story ) {
      $line =~ s/\[([A-Za-z-]+)\]/Replace($1)/eg;
  }
[download]

This replaces each bracketed work with whatever Replace() returns for that word. The /eg modifiers on the regular expression say to replace whatever is found with the results of executing (/e) the expression on the right-hand side, and to repeat this match "globally" (/g) across the target string. And, because $line is "aliased" to each row in @story, you don't need to make a second pass through @story.

Then, to print the story

  print @story;
[download]

Now, you've localized all of the interesting work to

  sub Replace {
      my $word = shift;
      ...
  }
[download]

which I'll leave as an exercise.

One (perhaps main) reason your script didn't work as intended can be found on this line:

       if ($word =~ /^\[/ && $word =~ /$\]/)
[download]

I think you're trying to use the dollar sign in the second regex to mean "if the word ends with a close-square-bracket", but since the dollar sign there is supposed to mean "end of string", you're supposed to place it at the end of the pattern, like this:

       if ($word =~ /^\[/ && $word =~ /\]$/)
[download]

But as others have pointed out, there is room for quite a few other improvements to the code...