The maximum voltage has more to do with the package dimensions than blow-up issues - if the maximum voltage across the device is exceeded it may arc across the terminals effectively bypassing the device.
Resistors can fail both open and short depending on the nature of the stress exposed to it.
A typical multimeter in voltage mode is tens of megohms. Blowing a resistor while measuring its voltage will likely only damage the probes. If you're measuring current, it will depend on the meter. Is it fused? If so, the fuse should protect the meter.
The "critical point" for a resistor is solely based on its temperature. If it gets too hot, the internal structure breaks down and the part dies.
If you have some sort of current-limited DC source, set it to 5mA or so and use it directly to test for the zener effect in your unlabelled diodes. Otherwise, use the highest resistor you can find to keep the current to a minimum (recognize that the zener voltage varies with load current) and test away, reducing gradually once you get an idea of the approximate voltage. Keep the total power in the zener below a few hundred milliwatts for safety's sake (if you don't know what power they can handle).