What version of HTML::TokeParser and HTML::Parser do you have? This works:

my $html = '<font class="copy">Name<br />Address<br />Country</font>';

my $p2 = new HTML::TokeParser(\$html);
while (my $token = $p2->get_tag("font"))
{
    my $text = $p2->get_trimmed_text("/font");
    print $text;
}
[download]

That prints Name Address Country.

Also note that HTML::Parser parses texts in 512K chunks. Is there a possibility that your actual program is hitting this barrier? See the $parser->unbroken_text method in the HTML::Parser docs.