My back entryway has this closet that kinda sticks out. I would rather have an entryway bench there instead. Just from glancing at this photo, do you think this is load bearing? Directly underneath in the basement is the furnace, and directly above is the guest room. Neither of which seem to have any structural support in the same area.
We can't tell with 100% certainty. It's very unlikely to be load-bearing. At a stretch ...the builder ran out of joists but had some that were 2 feet too short, so built the closet, used the shorter joists, and added some two-foot ones across the closet. I'm trying too hard right? But if you want to be 100% sure, since your intention is to demolish the thing and you'll have to patch and paint anyway, start by making a hole in the closet ceiling and looking to see if my silly imaginary scenario is fact. If it is, just patch up the closet ceiling crudely; it doesn't matter if it doesn't look perfect. Otherwise keep smashing!