The clocks are fairly simple and I really like them becuase when you gut them they look like clocks with a movement !
There are usually two things that stop them working.
1: as has been mentioned they are electo mechanical, they work using a small solonoid, when it sees power it fires and shoots the winder round, there is a spring which then assists the movement back to where it started, at the end of this is half a set of points, as it reaches the end of it's travel the points meet the other half, as this happens the solnoid activates and shoots the mech back round for another go ! simple and great to watch work!! The points rarely burn out, as Dave says they oxidize and eventually the contact is lost. To solve this part of the problem simply get a tiny piece of 1000 grit wet and dry, fold in half and rub up the points until they are shining and smooth, be very careful not to bend anything as it is a fragile piece, once you have renewed the ponts surface they should be good for a few years, this leads us to.....
2: The mechanical part, you need to think of this part as a standard clock, to check it is working manually open the points to around half the distance the solonoid would fire them, hold the clock as it would be in the car and observe, the clock should move slowly and steadily back to the points closed position without slowing as the second hand climbs towards 12 O'clock chances are it won't ! the whole movement runs on tiny brass bearings with steel pins running into them, over time (

) the brass wears ever so slightly and the bearings start to bind and slow the ,ovement and eventually the resistance stops the second hand climbing past 6 O'clock, this is a simple fix as well! VERY carefully damp the whole movement with WD40 keep it away from the face at all costs and try and keep it of the pin that goes to the fingers, leave it a few minutes then blow the excess off and go through the manual test again, it should now be very happy and climb enthusiastically towards 12 O'clock, if not repeat.
Now apply power to the clock and check the solonoid fires and the clock runs smoothly for a few cycles the re fit to the car, job done.
Note: B body tic toc tachs can be removed from the dash by squirming under the dash and removing the wires and the three screws that hold it in place (remember to remove the adjuster knob first!)