Well firstly avast isn't a firewall so isn't designed to block, the Web Shield only monitors inbound http (port 80) traffic and alerts to infected content but doesn't block unless you set it up to do so in the URL Blocking of Web Shield. What is your firewall ?
I didn't know anything about Teamspeak other than a quick google search, which tells me it is VOIP and I don't believe avast monitors that traffic as it uses a different http port when you use the web-interface.
This would appear to be something other than avast as your test with avast disabled would tend to confirm. I don't know which elements of avast you disabled/terminated (paused may not be enough), but the Network Shield and Web Shield are the only two providers that relate in any way to internet traffic.
I assume this is the FAQ you were looking at,
http://www.goteamspeak.com/index.php?page=faq&id=4&item=16