The problem I can see in your code is that you are not waiting for the last children to exit before writing to the log file. I would write is as follows:

#!/usr/bin/perl

use strict;
use warnings;

use LWP::UserAgent;
use Proc::Queue size => 3, qw(run_back waitpids);

my $starttime = time;

my $user_agent = LWP::UserAgent->new;
$user_agent->timeout(30);

my @urlstoget = qw( http://foo.com http://bar.com http://etc.com http:
+//and-so-on.com );

my @pids;
for my $url (@urlstoget) {

    push @pids, run_back {
        my $request = HTTP::Request->new('GET', $url);
        my $response = $user_agent->request($request);
        my $dbh = connectdb('listeningpost');
        processresponse($response)
    };
}

waitpids(@pids);

my $elapsed = time - $starttime;

open LOG, ">", "getitemshtmlog.txt" or die "unable to open log file";

print LOG "$n urls in $elapsed seconds, ",$elapsed/$n," sec per url\n"
+;
[download]

If this version doesn't work yet, it could be a bug on Proc::Queue, in that case send the author (=> me) a bug report with your full script or post it here.