Burning limit for a dc motor
An ideal motor driver will monitor the motor current and react to it much faster than you can manually react. Manual operation may cause some damage to the the commutator every time you energize the motor whether it is stalled or not. If you energize the motor momentarily while stalled repeatedly, the motor winding insulation may be damaged within a few seconds. It may only take a few minutes of that kind of operation to experience shorted winding turns. If you can obtain detailed specifications for the exact motor there may be sufficient information to provide a numerical estimate. Any experimentation of this kind is likely to shorten the life of the motor considerably.
A thermal switch on the motor frame is unlikely to prevent motor failure under the conditions described. It would need to be embedded deep in the rotor windings.