Do bees sting each other if they get into a fight?

Battle to the death if a bee from another hive tries to enter a different hive, especially a new queen bee. If not accepted she gets balled by the workers.

https://www.youtube.com/watch?v=DobxqhczJyQ

Yes; they might get into a fight if there are two queens, or if one bee colony invades another hive.  But this is rare.

It is more common that they will sting a wasp that invades the hive.  

If there are two queens in a hive they may fight to the death by repeatedly stinging each other. Workers are generally docile with each other.