I added an extra set of curly braces and that works. Evidently with this syntax, you have to "help" things out a little bit to get the value of "fh" otherwise it doesn't parse right. So this is how to make it work. Exactly why this is necessary is a syntax issue that I don't know the answer to.
#!/usr/bin/perl -w
use strict;
use FileHandle;
my $hashref->{"fh"} = FileHandle->new(">test");
print {$hashref->{"fh"}} "Hello\n";