Can a room "don" posters?

I would say no, not because "don" is only for clothing, but because "don" doesn't mean "wear" it means "put on" - an activity, not a state.

Merriam-Webster shows 'don' to be used with clothes only.

'Adorned' can be used, but it has a somewhat more specific meaning as can be seen in this definition (again from M-W): "to enhance the appearance of especially with beautiful objects". With beauty generally being considered as being in the eye of the beholder, the phrase also carries a certain judgement (lacking a better word for this in my vocabulary).

Depending on the number of posters, 'covered' could be used instead. If you are not picky about the exact phrase but are looking for a way to convey that there are a lot of posters in the room / on the walls, you could also try to rephrase the sentence to get the message across.

The phrase donned with appears to be used occasionally in reference to things, other than clothing, that can be said to be "worn," metaphorically speaking. A Google search reveals such instances as a cake "donned with...Sour Cream Chocolate Frosting" and an all-season tent "donned with sturdier poles and heavier fabrics than three-season models."

Depends on whether you want to reflect how most people use the word (for clothes only) or you want to be imaginative, poetic, and metaphoric.

I'd say go for it - everyone will understand you, and it's an interesting image. But at least be aware that you are being creative - this is invention, not convention. That doesn't make it bad or inappropriate, for most contexts.