What are the potential dangers of not removing the tabs on a split receptacle?
With a 120/240V single split-phase system, there are two possible outcomes.
Separate legs
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.
Same leg
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.