Question: If I feed a duplex receptacle from two separate branch circuits, I'm supposed to remove the jumper tabs between the receptacles. But what would happen if I didn't remove the tabs?

Answer: 
There are two main hazards. 

Overloads can occur, as detailed in other answers. Neutral wires and the loads themselves can easily be overloaded, causing all kinds of problems - even a fire. 

The other hazard is that turning off the breaker for a circuit will not have the intended / expected effect of de-energized the loads and receptacles on that circuit. This could be dangerous, and will be confusing, for the person trying to turn off that circuit to do maintenance or etc., or for situations where the breaker is used as a switch to turn things off when not in use.
If you add two live 15 amp lines to one outlet you will blow the breakers if not the main.