Perl has the warn function as a shortcut for print STDERR, so here is a different approach which uses a __WARN__ handler:

use strict;
use warnings;

$SIG{'__WARN__'} = \&whandler;
my ($Errors);

warn "Going to call foo()\n";
foo();

print "\nErrors:\n$Errors";


sub foo
{
    # this could be a loop that takes some time to finish.

    warn "Something to worry about\n";
    warn "Something else to worry about\n";
    bar();
}

sub bar { warn "Darn it.\n" }

sub whandler
{
    my $msg = shift;
    my $caller = (caller(1))[3] || '';
    if($caller eq 'main::foo') {
        $Errors .= $msg;
    }
    else { warn $msg }
}
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Here only calls to warn made directly by foo() will be stored.