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?
With a 120/240V single split-phase system, there are two possible outcomes.
If the branch circuits feeding the device are on separate legs of the service, then the tab will be creating a direct short-circuit between the legs.
This will cause a high current through the circuits, which should trip one of the breakers fairly quickly.
If the receptacle is fed by branch circuits on the same leg of the service, the danger is a bit more subtle. Initially everything works fine, no sparks, no tripped breakers. In fact, this situation can go unnoticed indefinitely, and never cause a problem.
The problem only occurs, if you try to draw more current than the individual branch circuit ratings. In the example below, the branch circuit protection for each circuit is 15 amperes.
If you put a load on the circuits, you'll find that the current will take all available paths. This means that the current will actually be divided between the circuits, so that each will see half the current that's flowing through the load. The yellow highlighting below, depicts the electricity's path through the circuits.
Still, there's no problem here. As long as the load does not draw more than the circuit rating, everything chugs along just fine.
The effects of this mistake only become evident, if the load draws more than the circuit rating. If the load in our example draws 30 amperes, you'll start to see the problem.
If you measure the current flow at various points along the circuit, the problem becomes obvious. Each branch circuit only sees 15 amperes, so neither breaker should trip. However, the receptacle will have to handle the full load current, which is well beyond the rated current. The receptacle will heat up and fail, potentially starting a fire as it does.
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.
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!
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.