I saw a picture of a potted lemon tree with Irish moss covering the soil and it looked really nice.

Are there any drawbacks to putting moss around your potted tree? I'd like to try it. Will the moss compete with the tree for nutrients? Do the risks outweigh the benefits of a nice looking tree?

I would be worried about the dampness, rather than nutrients.
Lush green moss thrives in damp conditions. It also tends to hold water to itself rather like a sponge. Dampness close to the stem of a potted citrus is considered bad, because there is a risk of foot-rot problems.