First off you will want to open your file with >> not >. >> appends to the file, > writes over the file:

open (FILE, ">>$filename") or die "trying";
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Secondly, if you want to actually write to that file handle, you need to specify the file handle when you print:

print FILE "$movie\n";
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Also, you might want to wrap the whole grabbing of user input around a while loop - otherwise your @movies array is totally unecessary.

Lastly, but very important - don't quote bare variables:

push @movies, "$line"; # bad habit to get into
push @movies, $line;   # much better, what if $line was a reference?
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jeffa

L-LL-L--L-LL-L--L-LL-L--
-R--R-RR-R--R-RR-R--R-RR
F--F--F--F--F--F--F--F--
(the triplet paradiddle)