Assuming that your file is always balanced and there are no qouted { or }, it can be made much simpler. (I didnt bother looking into your regex's or the Text::Balanced more than breifly, so bare over with me if this is not what you want)

#!/usr/bin/perl -w
use strict;

#input
print "Name of file to be inputted? :";
$infile=<>;
print "What should the output file be named? The file will be automati
+cally created. :";
$outfile=<>;
chomp $infile;
chomp $outfile; ##<<<<<<<<<You forgot this one
open INFILE, "<$infile";

#output
open FILEOUT, ">$outfile";
my @stack;
while(<INFILE>) {
  my @op = ();
  my $len = length($_);
  my $minsta = 0;
  my $minsto = 0;
  my $lastpos = 0;
  while(true) {
    $minsta = index $_,'{',$minsta;
    $minsto = index $_,'}',$minsto;
    last if ($len == $lastpos || ($minsta == -1 && $minsto == -1));
    if ($minsta < $minsto) {
      my $nextsp = index $_,' ',$minsta;
      $nextsp = length($_) if ($nextsp == -1);
      my $stack = substr $_,$minsta+1,$nextsp-$minsta;
      push @op, (substr $_,$lastpos,$minsta-$lastpos), '<',$stack,'>';
      push @stack, $stack; 
      $lastpos = $minsta+1;
     } else {
       my $stack = pop @stack;
       die "Not balanced" unless (defined $stack);
       push @op, (substr $_,$lastpos,$minsto-$lastpos), '</',$stack,'>
+';
      $lastpos = $minsto+1;
     }
   }
   print OUTFILE join "",@op,(substr $_,$lastpos,$len-$lastpos);
}
close INFILE;
close FILEOUT;
die "Not balanced" if (@stack);
print "\nYour result is stored in file $outfile\nGoodbye.\n";
[download]

I may have some offby 1 errors, solve them if you find it usefull...

This will use almost no memory and prob do the same jobs a LOT faster.

Update:
Well, seems tilly had the same idea as I, and dinner interrupted me so he got it first ;)

T I M T O W T D I