I have an old, smaller tree stand. I noticed that it has starting to rust. Will this hurt my new live tree? I'm thinking, yes, but not sure.
Rusting Old Tree Stand
If by "live" tree you mean a fresh-cut tree (and I'm making the assumption that you're talking about Christmas trees,) as long as the rusted stand is still water tight, the rust will probably not affect the tree. It it's severely rusted, flaking off large amounts of rust all over, you'll probably be happier with a new tree stand.
If by live tree you mean a tree with roots planted in soil, usually in a pot (this is the usual meaning of live tree,) and your tree stand is some sort of support contraption, rust should not cause any problem for the live tree. Here's an article that you might find helpful: https://www.gardeningknowhow.com/ornamental/trees/christmas-trees/christmas-tree-care-caring-for-a-live-christmas-tree-in-your-home.htm