Does planting hosta plants around a tree kill the tree?
No, not as long as the tree can still get plenty of water. Hostas don't have extensive root systems that will entangle themselves in the root system of the tree. Having said that, you might to make sure whatever tree you choose to plant them around is mature enough that the roots are already well-established.
Here are some tips for growing hostas: https://www.gardeningknowhow.com/ornamental/foliage/hosta/growing-hosta-plants.htm