Darn... Okay, how about opening STDERR to a file, then closing the file and deleting it? It's not pretty, but it may work.

use Net::SMTP;
my $tmpfile = "$ENV{TMPDIR}/stmp.$$";   # Or use a module like File::T
+emp

open(OLDERR, ">&STDERR");
open(STDERR, ">$tmpfile");

eval { my $smtp = Net::SMTP->new('relay.utah-inter.net') };

close(STDERR);
open(STDERR, ">&OLDERR");

unlink($tmpfile);

die $@ if $@;
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And here's a link to the File::Temp module.