For a non-lethal run, do I need to keep bodies off the ground?

Yes; in Dishonored 2, rats can still eat unconscious bodies, which will count as a kill in your mission stats (in the pause menu), just like in the first game. In fact, this time rats will eat bodies even if there are only 4 or 5 of them running around (which, in Karnaca, there usually are as you've implied from your rat plague comment).

I've noticed, however, that this does not apply to

rats eating unconscious Nest Keepers,

and it's unclear to me if that is intentional or an oversight.

If there are only one or two rats, though, it shouldn't be a problem. Not all groups of 4 or 5 rats eat bodies in my experience, but it was quite a rude surprise to find out that some were doing just that.