If I check the boxes to enable twitter for profiles or groups in journal settings additional options for status updates, I can't save them.  Also, the popup window won't go away.  I tried both IE and Chrome.  I'm running the latest AS and DNN.

Moved to proper forum.

Please make sure there aren't any javascript errors on the page. Also check the event viewer.

No javascript errors and no event logs.

Even if I manually change the entries in the database, the checkboxes don't show up checked when I open the popup. If I save the page, the database entries get changed back to false.

I did however discover that to close the popup you need to click the ellipses again. Would be nice to have a close or X button.

Sounds like your site is running under medium trust or something is preventing you from connecting to twitter. When Active Social can't connect to twitter for one reason or another it will disable those settings automatically.

What trust should it be running in?

Full or modified trust that allows for WebRequest permission and remote web connections.

I just checked and my site is running in Full Trust mode.