I am not sure if I am answering the correct question.
1. If nothing is connected to the generator the circuit is open (i.e. not connected), the electrons won't flow. No electricity will be generated. The generator will run at the idle speed (and will use the minimum amount of fuel). The energy from the fuel will be turned to heat... but since no electricity is being generated it will be a limited amount of heat that dissipates safely.
Your image of the electrons piling up is incorrect. Electricity doesn't work that way. A slightly better metaphor is when you turn off a faucet there is still water pressure, but the water doesn't flow. The water pressure running the faucet stays the same... it doesn't go up continually just because the water isn't running.
2. If you short the circuit, it could be dangerous. This means you connect one pin of the the generator to the other allowing the electrons to flow in the circuit with no load. Generators generally have a fuse or circuit breaker to break the circuit when this happens. If you didn't have this type of safety feature, it would overheat in a dangerous way.
To answer a possible follow up question... the more load you put on the generator, the harder the generator is to turn (this is Lenz's law). The generator will burn more fuel the greater load you put on it.