If this happens to a race engine running at full chat it could have some nasty consequences. When it happens to a standard engine in normal use it usually does not cause any other damage. If an intake valve does not open it just runs on three cylinders with some unusual vacuum in the cylinder on three of the four strokes.

If an exhaust valve doesn't open it makes a lot more noise. After the normal power stroke the exhaust gas cannot escape, so it is re-compressed to high pressure during the exhaust stroke. When the intake valve opens the exhaust gas escapes suddenly back through the intake manifold. That makes a big bang with every second revolution of the crankshaft, and it can really screw up the intake air flow as well as possibly igniting the intake vapor (which might blow out the air cleaners).

If you encounter this problem, remove the broken piece and the pushrod. If an exhaust rocker has broken, then also remove the intake pushrod on same cylinder. Then start it up and adjust for appropriate idle speed, and you can drive it home running on three cylinders.

The repair is usually as simple as replacing the broken rocker arm, but you should also do a compression check to be sure it does not have a bent valve.