You could argue that your use of overloading is too confusing.

That warning is generated at compile-time, and is correct for non-overloaded invocants (>> returns a new value, and doesn't modify its parameters), so I'd say the behaviour is expected, but annoying in this case.

You can disable the warning by using

no warnings 'void';

By the way, stringification gives you the same warning in similar cicumstances:

#!/usr/bin/perl -w
use strict;
my $foo = Foo->new;
"$foo";

package Foo;
use strict;
use overload '""' => 'stringy';
sub new {bless {}, __PACKAGE__}
sub stringy { print "OK!\n"}
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Useless use of string in void context at test.pl line 4.
OK!
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