Hi Leah,
The error indicates a timeout when selecting unprocessed task rows from the database so my first thought was maybe there are lots of rows but this does not seem to be the case. You could try running Exec mp_TaskQueue_SelectIncomplete from query analyzer and see if it runs fast or slow from there.
I notice you are using a connection string with integrated security which means it is using a windows user not a sql user. If possible I recommend use a sql user. The tasks run on a background thread so I'm wondering if the background thread doesn't have the same windows user context as threads that execute web requests then maybe it can't connect to the db.
You could disable the TaskQueue with <add key="DisableTaskQueue" value="true" /> but then your search index won't get populated, newsletters won't send and lots of other little tasks that run from our WebTaskManager would no longer do their jobs.
Hope it helps,
Joe