I'd like to opt out of getting gifts from people for the duration of Lumi, but I am unsure how to do that. I have removed the tree section from my profile page, but I am unaware of how to block all gifts. Is there something I can do aside from asking people not to give me things?

No, I don't believe there's an option to opt out completely. You can turn off accepting gifts from non-friends on this page but I think that might only apply to the gift center, not trees.

Good luck with that, Subeta's not set up for it.

If someone is determined they can give you stuff under your non-tree by using URL tricks. These gifts will open up on 25 December no matter what you do.

For Arctic Frost, you'd have to unfriend everyone and choose don't receive gifts from non-friends.