Torque output from an electric motor, when a force is applied to it's movemet
If a force is applied against rotation, (induction motor for example) slip will increase, causing the current will increase, resulting in a greater magnetic force in order to drive the load.

If the same motor is loaded down to such a degree that the rotor cannot maintain near synchronous (minus slip) speed, and there is no overload device, then yes, it will burn up as the current will only be limited by the winding resistance without any counter EMF.  You don't have to get it to zero speed to burn it up, just to the point that it falls below the design synchronous speed minus the slip.

The torque is a different beast.  That depends on the motor and the method of control.

In a squirrel cage induction AC motor, across the line, maximum possible deliverable torque occurs at the design speed (synchronous minus slip).

The maximum deliverable torque is typically about 200% of the rated torque.  Induction motors are also typically designed for at least 110% starting torque.  The torque will drop off to zero at synchronous speed.

With a squirrel cage induction AC motor on a typical vector drive nowadays, the torque can be maintained pretty much constant over the entire speed range without damage to the motor (though a blower might be required).

A DC motor from zero to base rated speed will operate in a constant torque mode, and full torque can be delivered from zero speed.  Again, a blower is required.  Past base speed (field weakening), the motor will be operating in a constant horsepower mode; the deliverable torque will drop off as the speed increases above base in a linear relationship.