I have a PhD in Physics and my thesis was, in part, related to wormholes, so here it goes. (Squark covered most of your question already, though.)

If something falls into a black hole, it increases the black hole mass. If something escapes a black hole (such as Hawking radiation), it decreases the black hole mass. Same with white holes. A wormhole is basically two black/white holes connected by a throat. One pass through the portal would increase the mass of the entrance and decrease the mass of the exit by the mass of the passing object.

A portal with two ends having opposite masses would behave rather strangely: they sort of repel (the equivalent of Newton's law of gravity), but the gravitational force acting on the negative-mass end propels it toward the positive-mass end. As a result, the portal as a whole will tend to accelerate toward the positive end (entrance) and fly away, albeit rather slowly.

In addition, due to momentum and angular momentum conservation, the portal will start spinning to counteract the motion of the passing object.