First, elevators have counterweights, so for some mid range load there is no net potential energy change in the elevator going up or down. However, a maximally loaded elevator probably has more potential energy than the fixed counterweights, so your question still stands.
In any case, the friction in the pulleys, motor losses, etc, all have to be overcome before there is any net gain. I don't know if this is typically the case when the elevator is fully loaded or not, but let's say it is.
At this point, it's at least theoretically possible to run the motor in such a way that it returns power (acts as a generator) while regulating the elevator speed. Whether this is actually done, I don't know. The few elevator systems I am familiar with don't do this, but these are older. If this is done, I expect it to be more prevelant recently where energy costs are higher and advanced motor control more accessible.
One system I know of works on hydraulics. It is only a 3 story building. There is a hole in the ground below the elevator that the piston fits into. I know that system is counter weighted so there is always net down weight on the piston. To go up, a electric motor drives a pump to raise the piston. To go down, it simply opens a valve to let the hydraulic fluid in the piston back onto the reservoir tank. In that case, the potential energy of the elevator heats the hydraulic fluid a bit as it passes thru the valve.