Are there any risks using a non-microwavable plastic plate in the microwave? We have some plates we bought from Target a few years ago that we frequently use in the microwave.  These plates are listed as not being microwave safe.

Does that mean that they could cause health issues if we use them in the microwave, or does that just mean that the microwave could ruin the plates?
Yes, there are risks. Avoiding those risks is the entire point of the microwave-safe labeling.

When plastic is marked as microwave-safe (in the US), it means that the material has been tested and found not to release harmful chemicals when heated, by FDA guidelines. This is a stronger assertion than just that it doesn't melt. Something that's unsafe in the microwave may not show it visibly; it may leach chemicals without melting.

It's entirely possible that something not marked as microwave-safe may actually be safe, and just not be tested. But you have no way to know, and you should assume that anything unmarked could potentially be harmful.