Are mushrooms healthy or bad for finger lime?

They shouldn't hurt it.Mushrooms are growing because spores made it into your potting mix and the conditions are right. It has more to do with the soil vs the plant, but some say it can be beneficial to the plant. At the very least it shouldn't cause harm and you can remove the mushrooms if you don't like them. If you want any to get rid of them, repot into a new medium.