Autopilots used in piston GA usually do not have throttle control. They only manage the control surfaces. However to trim an aircraft one needs to play on both throttle setting (and more physically, thrust) and control surface deflections (aerodynamic forces). What happens if the autopilot cannot trim the aircraft due to the propulsion settings? Is there any alert from the autopilot for the pilot?
It all depends on the autopilot, but generally no. If for example, you tell the autopilot to climb at 300 fpm, it will keep adjusting pitch to do just that. As you climb higher (without changing the power) the autopilot will keep increasing pitch to maintain the set vertical speed. Eventually the airplane will get too slow because of the increased pitch and can even stall. Some autopilots have pitch limits or monitor the stall inputs and automatically disconnect before it gets to this point, but others don't. It is very important to know your autopilot and to monitor it at all times to make sure that it is doing what you want.