The way the Constitution is written, it appears that the House would have to pass any revenue increase bills.
The first sentence reads as follows:
All bills for raising revenue shall originate in the House of Representatives; but the Senate may propose or concur with amendments as on other Bills.
That is about a clear as you can get on that topic. As for what can be done, the answer is not a whole lot. But that is the current aim of the GOP.
Their goal is to keep the Dems from being able to accomplish much of anything, and then retake control in subsequent elections. It turns out that they have realized that many voters: a) are stupid, b) have short memories or c) a combination of both of the above.
As such, most of us have forgotten that the reason the Bush tax cuts were extended was to get a vote on the extension of unemployment benefits even brought to the floor. And that happened only eight months ago.
They have managed to muddy the debate just enough to confuse those who aren't paying attention.
Hence the current debate about the debt ceiling even though it has routinely been raised under GOP administrations.