I recommend Filter::Handle. I think it will do exactly what you want. I used it for the very same purpose -- to log the output a script without changing print statements, etc. Here's a sample script.

#!/usr/local/bin/perl

use warnings;
use strict;
use Filter::Handle qw/subs/;

#open the logfile
open (LOGFILE,">>test.log") || die "open: $!";

#set up a sub to filter our output
#print @_ to LOGFILE then also print it to STDERR
Filter \*STDERR,
    sub { local $_ = "@_"; print LOGFILE scalar(localtime), ": $_"; $_
+};

#test it out by printing something to STDERR
print STDERR "hello world\n";

#turn off filtering
UnFilter \*STDERR;

close LOGFILE || die "close: $!";
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When run, this produces the following output:

/home/rhet/misc> ./filter.pl
hello world
/home/rhet/misc> cat test.log
Thu Sep  6 11:46:24 2001: hello world
/home/rhet/misc>
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