Burnt Windings?
If the windings are burnt, a visual check may reveal discoloration and/or visible charing, however not every portion of the windings may be visible.
However, a better check would be a continuity check of the armature and field coil.
The field coil, being basically a wire wrapped around a plate, is checked like this:
Using a multimeter set to read resistance (Measured in Ohms), and the leads properly hooked up, check the continuity of the wire, it should have very low resistance, usually in the order of zero to a few ohms, as it is a wire. A very high resistance, for example hundreds of Ohms or above, would indicate an open circuit, i.e. a broken wire.
Also check the resistance between the field coil wire and the field plates. This should indicate a very, very high reading, typically infinity, or whatever the full scale reading is on the type of meter you use.
If there is a low/no reading between the wire and the plates, it means that there is a short between the wire and the plates, which would prevent the fild coil from building a magnetic field, which would yield a motor that will not run.
As far as the armature is concerned, check the resistance between commutator segments. Each reading should be zero/very low. Also, at this time it may be a good idea to check each of the segments for pitting, and looseness.
Check the resistance between the commutator segments and the shaft. This should be very very hich/infinate. A low reading would indicate a damaged armature. It may be possible to fix said damage at home, but unlikely, and is usually a PAIN.
While the motor is apart, check the condition of the brushes, and the brush holders. If there is insufficent pressure on the brushes, the motor will not run or at least run terribly.
When reassembling the motor, ensure that the motor turns easily and smoothly, if there is any binding, it can yield overheating which CAN burn out windings if left unchecked.
If you do not