Is it okay to cover food in the fridge with paper plates? If I had some chicken and pasta on a ceramic plate and wanted to save the rest for later, when I put it in the fridge, is it fine to just cover it with a paper plate or should you use saran wrap?

Ideally, the paper plate is much more convenient and quicker to use, but I have concerns that the smell and taste may leak to other foods in the fridge.

How safe is it to use a paper plate to cover foods like this. If so, for how long? Usually I eat the food within the next day or two but am wondering how quick the process may be.
Ideally you should get airtight containers, but failing that, plastic wrap is indeed a lot better than just covering with a plate.

As you say, you'll tend to get more smells mixing around in the fridge if things aren't airtight, and it's something that can kind of build up over time, with the interior of the fridge just taking on a mix of all the smells. (Fridges can eventually smell weird even if you're not doing this.)

It's also not generally as good for the food; it'll dry out faster. Might not be an issue for everything, but sometimes it can be pretty obvious. It may also get contaminated a little more easily and spoil faster, but that's a little random, hard to say anything specific about time periods.