I've noticed that mobs don't just walk off of cliffs on their own accord. They stop when they sense a cliff (unless they are chasing you). How high does a cliff have to be for them not to walk off of it?
A pig will voluntarily walk off an edge 3 blocks high, but not an edge 4 blocks high.
To investigate this, I started with @xfs's proof that a pig will not jump off blocks in a 9-patch arrangement if the drop is too high. I then constructed the following apparatus to find the maximum distance a pig would voluntarily jump off:
As you can see, a 2 block drop is fine
A 3 block drop is also fine
At a 4 block drop the pigs no longer jump off voluntarily
Conclusion: The highest drop pigs will drop off voluntarily is 3 blocks
From what I have seen, most mobs wont wander off a two block high cliff unless they have seen you. It's very difficult to say since if you get too close their behavior changes. It could also be different on a mob by mob basis. I have read that cows will ignore cliffs entirely and plummet to their doom!
If you are holding wheat in your hand, a mob will follow you off of any drop, and will happily drop off a cliff while circling you for the wheat.
If you are leading a pig wearing a saddle and carrying another player, this is a very easy way to get the related achievement - lead the pig to the cliff, and stand at the edge. The pig will circle around you and drop off.