Why are you passing typeglobs? There's no reason to use typeglobs here, just pass the lexical handle as a scalar. Plenty of older tutorials say to pass filehandles as globs, but that only applies to global filehandles generated when you do stuff like: open( FOO, ">/bar/baz" ). It's best to use a 3 argument version of open and a lexical handle.

#!/usr/bin/perl
open(my $handle1, "<", "/tmp/1") or die "Input file does not exist.\n"
+;

# Case 2 -- fails
GetFileContents($handle1);

close($handle1);

sub GetFileContents
{
  my $foo = shift;
  print "Function hash:\n";
  {
    local($/);
    print <$foo>;
  }
}
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On a side note, I like to use the lower precedence or for error detection, becuase it won't mess with my lists if I leave off parens. See perlop section on C-Style Logical Or for more info.

Update: Clarified globs vs typeglobs. Also, it's worth pointing out that using a lexical handle and not a typeglob doesn't resolve the issue with the <$hash{lexical_handle}> construct. Others have covered that issue.