2011 HHR 2.2 have replaced battery, alternator. The motor now starts but starter will not disengage.
Yes, the starter could catch fire or it could be completely destroyed if it remains engaged while the engine is running.
Check to make sure that the ignition key is returning by itself to the RUN position after you release the key from the START position. If it is not, turn the key back to RUN after the engine starts, and the starter should disengage. In that case, the ignition switch needs to be repaired or replaced. (Happened to me on an Audi.)
The ECU controls the starter on this vehicle, moving the key to start initiates a start sequence - which can continue after the key is released, and the key being in or moved to start is ignored once the engine starts, so it's not the switch. Most likely the starter motor is damaged, either the solenoid is sticking mechanically, or the contacts are welding closed and holding the solenoid and drive in, or the pinion is sticking on its shaft - there are bronze bushings inside the pinion itself, and the clutch behind it runs on a helical spline, both of these can get gummed up and hold the pinion in mesh.
There is a risk of the armature, particularly the commutator bursting if it is held in mesh while the engine is running, which can shred many of the insulation parts in the motor, so there is some risk of fire. The one-way clutch behind the pinion is meant to prevent the armature from being oversped, but it is only effective for a short time.
One final possibility is that the starter relay in the fusebox can also have contacts sticking, this is worth checking out before looking at the motor.