The loop over the %FORM hash keys is really confusing. I think you don't need it. From your description you want to check the lines from the input file against the hash keys, but you never do that. I think you need something like that (not tested):

open OLD, $file or die "Can't open $file:$!\n";
open NEW, ">$file.new" or die "Can't open temp file $file.new:
+ $!\n";
my $found =0;
while (<OLD>) {
    if ($FORM{$_}==1) {
        $found =1;
        next; 
    }
    print NEW; 
    }
}
close NEW;
close OLD;
if ($found) {
    rename ("$file.new", $file) or die "Can't rename new $file: $
+!\n";
} else {
    print "I didn't find !\n";
}
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What I changed:

• if ($FORM{$_}==1) this is checking if the line is a key in the %FORM hash and has a value of 1 (you might change it to  if(exists($FORM{$_})) if you need to check just existence).
• Closing the NEW and OLD filehandles.

You could look at File::Atomic if you run this concurrently.