How long can a submarine stay underway?

It depends on the propulsion type (Diesel-Electric or Nuclear). DE's are of course limited by fuel replenishment requirements as always, but today's boats are much more efficient than their WWII counterparts, and can typically stay at sea for many weeks.

Nuclear Submarines are a different animal altogether. Were it not for crew needs (food replenishment) and maintenance that can only be done in port, they could stay underway indefinitely until refueling was required, which is typically several years.