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: 
It's worse than tester101 says. Each circuit may not see half, there may be an imbalance of currents between the redundant paths. (Especially if one has a problem, such as being completely broken). Now how are the wires protected from overload? The hots have rather nice breakers on them, but the neutrals are not breaker protected! Nothing will detect an overload there. 

[![enter image description here](https://i.stack.imgur.com/YXHzz.png)](https://i.stack.imgur.com/YXHzz.png)

The opposite is an even bigger problem - suppose current flow does balance. One circuit has *other* outlets, and those are overloaded to 27A. They draw through this outlet, drawing a relatively balanced 14A from one breaker and 13A from the other. Whoops! 

[![enter image description here](https://i.stack.imgur.com/ESpL2.png)](https://i.stack.imgur.com/ESpL2.png)
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.