rename "$file_in", "$random_num"."$file_in" or warn "Couldn't rename f
+ile: $!\n";
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I would put the random number at the end of the file name. Could $file_in have any slashes in it? If it does, then the path you're trying to move it to won't exist.

Also, I would simplify your code by just directly checking to see if the file exists -- no need to iterate through the directory:

if (-e $file_in) {
    rename($file_in, "$file_in.$random_num");
}
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Of course, you probably want to check if "$file_in.$random_num" already exists:

my $f = $file_in;
while (-e $f) {
    $f = "$file_in." . int(rand(100000));
}
rename($file_in, $f);
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But this doesn't protect you against race conditions - like, what if two people upload the same file at the same time. To protect against that you'll need to find an identifier that uniquely identifies this upload.