What happens after a zener diode breaks down?
First, why will the zener fry?  You may have a 12V supply powering your regulator, but your supply must have a fuse or circuit breaker somewhere to be safe.  If this fuse or circuit breaker has a current rating which the zener can tolerate, then your PCB should be protected by the zener while the fuse shuts off, and no harm will come to the zener.  If the current rating is higher than the zener can tolerate, well, you've made a mistake in choosing a current limiter.

What is the function of the 6V zener?  If its function is to protect the 5V rail of your circuit, then I suggest that it's not going to do a very good job.  Many 5V components have a maximum input voltage of around 5.5V or 6V, so a 6.2V zener won't help much.  I'm not sure what your environment is like, but it's usually better to just shut everything down if your regulator fails than to try to let the zener run everything.  

One common use of a diode for protection from error conditions is to use a reverse-biased rectifier diode across the input terminals.  That way, if anyone connects the source backwards, the power will be dissipated in your designated diode.  Make sure that this diode's forward voltage is less than the maximum reverse voltage on the protected components.  In this configuration, when the source is plugged in backwards, nothing will work: The supply voltage will be at -0.7V.  Presumably, you'd notice that the power LED was not on or that the circuit was not functioning, and correct your error.

Second, no, you can't say whether the result will be a short or open circuit.  You've operated the device outside of the specifications, so anything could happen.