What happens to the Arduino if you don't use a resistor with an LED?
From page 411 of the ATmega328P data sheet, we have a graph of the output current vs voltage. 



We can see over the the 20mA range of output currents, we drop ~0.5V, giving an approximate internal resistance of 25Ω.

A typical LED has a forward voltage of about 2V. Therefore, we end up in the situation where we will drop 3V over the 25Ω internal resistance of the ATmega328P, giving an output current of 120mA.

I = (5-2)V / 25Ω = 120mA 

This is a simplification, but this 120mA is far larger than the recommended maximum current per pin of 40mA of the microcontroller, and far in excess of what a normal LED should take.

That said, most LEDs and the ATmega328P seem to survive this kind of abuse, especially for short periods. In general, if the pin continues to work after this is done, it should be fine.