Says someone who's never actually tried it. The biggest risk of burnout is from the use of constant voltage supplies, rather than constant current. With a "normal" power supply, a stalled motor will draw enough current to overheat quickly and catch fire, as @jremington says. Plus, of course, its cooling fan isn't running.
Any brushed DC motor* can be held stalled without harm provided the temperature rise is not excessive.** I don't know of any way of ascertaining that except by experiment, although there will be motors specifically made for the job, and the spec sheet will give the necessary information.
For what it's worth, I'm not actually advocating a stalled motor approach for this application. I simply wanted to offer it for consideration, bearing in mind that @kill_gil wanted a solution that was as simple as possible, preferably without using any additional sensors.
*AC induction motors can also be operated stalled. Three-way splitter valves in domestic heating systems (in Europe, at least) use a stalled motor to hold the valve open against a spring when energised, the spring closing the valve when power is removed from the motor.
**An electrical engineering lecturer told me this rather informal approach to finding the maximum safe current. With an ammeter in circuit, increase the load on the motor (or transformer, or other wound component) until it just starts to smoke. Read the current. Then halve it to find the maximum safe continuous current.
So there you go: the maximum continuous current is half the smoke current.
Sounds like it could be a good rule of thumb, but it's probably worth buying a couple of spare motors. Make sure they're cheap!