foreach my $file (@the_files) {move ($here, $there)}
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You should check for success. Always. Always!

foreach my $file (@the_files) {
    move ($here, $there) 
        or die "something weird happened moving '$here' to '$there': $
+!\n"
}
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foreach my $file (@the_files) {move ($here, $there)}
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foreach my $file (@the_files) {move ($here . '/' . $file, $there . '/'
+ . $file)}
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Starting after your %image_hash setup:

my $gifdir_name = "/home/tony/perl/pmbencor";
opendir my $gifdir_dh, $gifdir_name
    or die "Can't opendir '$gifdir_name': $!";
my @orig_files = grep { !/^\.\.?$/ } readdir $gifdir_dh;
closedir $gifdir_dh;

my $there = '/home/tony/perl/newdir';
mkdir $there, 0777  or die "Can't mkdir $there: $!";

my $key_pattern = join '|', map { quotemeta } keys %image_hash;

use File::Copy;  # this should really be at the top somewhere

foreach my $file (@orig_files) {
    my $new_name = $file;
    $new_name =~ s/($key_pattern)/$image_hash{$1}/g;
    move "$gifdir_name/$file", "$there/$new_name"
        or die "Can't move '$file' to '$new_name': $!";
}
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Notes:

• You may want to grep out all files that start with a dot, not just "." and "..". In that case, just use !/^\./.
• I don't think "$there" is a very good variable name (but I haven't changed it).
• Your original code treated the %image_hash keys as patterns (that is, a key of "^" would have matched anything), but my code treats them as literal strings. If you want the pattern behavior, change the quotemeta map to "map { "(?:$_)" }".
• Given "%image_hash = ( x => 'y', y => 'z' )", your code could have done two replacements on an "x", changing it to "z" (but not definitely). Mine will only do one replacement.
• If you have "%image_hash = ( abc => 'x', abcd => 'y' )", and the $file is "abcd", I'm not sure which will match. If that's a concern, you may want to go back to a replacement loop and sort according to length or something.
• Since your source and destination directories are on the same drive, you could use rename instead of move
• I haven't tested what I wrote.

foreach my $file (@the_files) {move ($file, $there)}
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foreach my $file (@the_files) {
    foreach $key (keys %image_hash) {
        $file=~ s/$key/$image_hash{$key}/g;    
    }
}
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You modify a local variable without doing anything with it.

He is not modifying a local variable, he is using an alias for the array elements. So it should have an effect.

Your image hash parsing code makes the regex engine work harder than necessary. I've reworked this bit to show several alternatives.

The problem is that /.* .*/ creates a lot of unnecessary backtracking. The regex engine is greedy, so it will scan the whole string and match it to the leading '.*'. Then it has to back track and remove one character at a time until it finds a space. If you write /[^ ]* .*/, then the engine can read until it finds a space, then move on the next match criterion, must be a space, match that and slurp in the rest of the string to match the final '.*'. I once read a great article that talked about this in detail, but I can't track it down.

my %image_hash = ( );

while (<IMAGEHASH>) {
    # Limited scope of $search/$replace
    # You could just use split to split on whitespace:
    my ($search, $replace) = split;

    # Or if you want to split on the first single space character, and
+ force only 2 items:
    # my ($search, $replace) = split ' ', $_, 2;

    # or you could use a regex:
    # my ($search, $replace) = m/([^ ]*) (.*)/);

    # Suppress uninitialized warnings.
    # check for non-zero length, 
    # although eq '' checks should be fine, too.
    {   no warnings 'uninitialized';
        if ( length $search and  length $replace ){
            warn "Found uplicate entry for: $search\n"
                if exists $image_hash{$searc};

            $image_hash{$search} = $replace;
        }
    }
}
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Warning: all code untested.

my @the_files = ('apple', 'pear', 'food');
my %image_hash = ('a'=>'z', 'o'=>'u', );

foreach my $file (@the_files) {
    my $newFile = $file;
    foreach my $key (keys %image_hash) {
        my $replaceVal = %image_hash->{$key};
        $newFile =~ s/$key/$replaceVal/g;
    }
    print "move $file, $newFile or die 'Bummer!' \n"; 
}
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move apple, zpple or die 'Bummer!' 
move pear, pezr or die 'Bummer!' 
move food, fuud or die 'Bummer!'
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If I use warnings and diagnostics, it also says,

Using a hash as a reference is deprecated
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(D deprecated) You tried to use a hash as a reference, as in %foo->{"bar"} or %$ref->{"hello"}. Versions of perl <= 5.6.1 used to allow this syntax, but shouldn't have. It is now deprecated, and will be removed in a future version.

Instead of "%image_hash->{$key}", say "$image_hash{$key}".