What you're describing is called bus contention. I've personally only seen this cause damage once and I was using '80's era technology at the time. Most modern microcontrollers have bus arbitration logic built-in so that this won't happen when they are the only master. But if you have multiple bus masters, this can certainly occur.
The most common symptoms would be glitching or incorrect values alternating with correct values very rapidly. You would easily be able to see this with a logic analyzer. It can be seen with an oscilloscope but I find that it's harder to identify as such.
With a modern microcontroller acting as a single bus master, this will rarely be an issue (assuming you let the micro handle OE# generation and don't force the pin yourself). With multiple bus masters, I usually put a small Programmable Logic Device (PLD) in the system and design a separate bus arbiter to ensure that this does not happen.