The House Republicans will be unable to get any legislation enacted or any old legislation repealed because of the veto. The President will be unable to get anyone to initiate any legislation for lack of House control. Government will grind to a halt. The next time anyone has to run, they will run on “everything good that has occurred is because of me (prove me wrong!); everything bad that happened on my watch is because they prevented me from doing what I wanted to do.” And everyone else will believe one of two versions of this, because in the realm of the unfalsifiable, one can believe what one likes.
Gradually, the economy will start to recover. Obama will get the credit for this by virtue of the same irrationality that has blamed him for the current condition of the economy. As a result, he will be re-elected. This will make Republicans angry.
It’s around then that one would expect the House Republicans to impeach him.