Yowch! You're probably hitting an OS limit on the number of file handles you can have open. If using one of the methods of reading the file into a hash doesn't run out of RAM, you'll want to use one of those. Otherwise, you'll have to modify the get_file_handle function to close some of its file handles when it's about to run out. Just as a quick off-the cuff thing, it might¹ be good enough to simply close *all* the file handles when you reach some predetermined limit. Something like (untested):

my $Max_FH=1000; # Maximum # file handles you want
my %FHList;      # Holds file handles we've opened so far
sub get_file_handle {
    my $key = shift;
    if (!exists $FHList{$key}) {
        if ($Max_FH <= keys %FHList) {
           close $FHList{$_} for keys %FHList;
           %FHList=();
        }
        open $FHList{$key}, '>', "output_$key.txt" or die $!;
    }
    return $FHList{$key};
}
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¹ Some workloads have a handful of commonly-used tags, and a mess of onesie/twosies. If that's the case, this will occasionally close and reopen commonly-used tags, but it'll clear out all the lesser-used ones. If the commonly-used values are common enough, the opens and closes will amortize to a small amount of overhead. If your workload has an evenly-distributed set of keys, then you'll need to make get_file_handle much smarter...