There's something which you are obviously missing. Let's see if I can explain this in such a way as the penny drops.

use strict;
use warnings;

$| = 1;
my $filename = 'foo.txt';
open (FH, '>', $filename) or die "cannot open $filename for writing: $
+!";
my $wasout = select FH;
print "In the file!\n"; # Here you print on FH which is open for writi
+ng. No problem.
close FH;

print "OMG, a warning!\n"; # Here you print on FH but it is closed so 
+a warning is shown

select $wasout;

print "Sweetness and light.\n"; # Here you print on STDOUT again - no 
+warning. Joy!
exit;
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The above code generates only 1 warning (intentionally as an illustration for you). See select for all the good stuff. When you run this code you will see this on the terminal:

print() on closed filehandle FH at sel.pl line 11.
Sweetness and light.
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and this in foo.txt:

In the file!
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Is it all clear now?

It's also worth noting that using a bareword (or other global) filehandle is rarely considered best practice in these modern times so that might be the next thing you could change.

Update: Fix in diagnostic for open: s/read/writ/ (thanks soonix for spotting).