If you run this SSCCE then you see where one problem is

#!perl
use strict;
my $map = {
  one  =>1, two  =>2, three=>3,
};

my ($regex) = map {qr /\b(?:$_)\b/ } join '|', map {quotemeta} keys %$
+map;
change_file('test.txt',$map,$regex);

sub change_file {
    my ( $fn, $map, $regex ) = @_;
   # print $fn;

    open ( my $fh, '<', $fn ) or die "Could not open file '$fn': $!";
    my $str = do { local $/; <$fh> };
    print $str;
    close $fh;
    my $num_replacements = $str =~ s/($regex)/$map->{$1}/ge;

    if ( $num_replacements ) {
        write_new_file( $fn, \$str );
    }
}

sub write_new_file {
    my ( $fn, $str ) = @_;

    open ( my $fh, '>', $fn ) or die "Could not open file '$fn': $!";
    print $fh $str; #<-- error here should be $$str
    close $fh;
}
<
#test.txt
__DATA__
one two
three four 
five
[download]

What do you mean by "renaming the contents of the files"?

A file has a name and has content. The rename system call changes the name, not the content.

If you want to change the content, you will need to rewrite the file using open and print.

Please go back to step one and describe in short, simple English sentences what you have, and what you want, and what steps you think you need to take to get from the one place to the other.

I think you're confusing rename with rewrite.