I've left it on once and noticed it at 1000 MSL and switched it off... I've heard leaving the APU on results in everything from it just flaming out to blowing up the tires to fire bells to compressor stalls (when switching onto #2 bleed in flight) etc. etc. I've never let it get that far and the most common one I hear is a flamout.
Actually in the FE school they taught that the first indication of the APU still being on after takeoff was the Wheel Well fire alarm would come on. So the first time this happened we all were concerned that even though the fire bell was ringing, none of the fire warning lights on the instrument panel were illuminated. Unfortunately my eager co-pilot silenced the bell just about as fast as the bell rang.
I remarked to the co-pilot that I kind of wished he had not done that, silence the bell, as we had an overheat/fire condition indicated somewhere. So trouble shooting was now rather limited. So I told him to recheck the lights, there was a chance that two bulbs had gone out at the same time, but not very likely. Just about then I started to think on just what could set off an fire alarm, but not light up anything on the instrument panel. Then it hit me, the APU.
So I kind of stood up in my seat, craned my head around so I could see the APU panel and sure enough, the APU fire light was on. So I told the FE to shut down the APU, so he grabed the start/stop switch and moved it to off. Of course the switch is disabled while airborne, so I had to tell him to pull the APU fire handle. He did and the light went out immediately.
Then he started arguing with me, he claimed that he did shut down the APU before takeoff and that the APU must have
self-generated a start.
Besides being a terrible FE, he wasn't much of a pilot either as it turned out.
Give me a professional Flight Engineer any day.