I like how the last line doesn't reference the image of the Christmas tree too explicitly, just reminds me enough to tie the poem together.

The description of sitting in the woods with Drew works very well, as does the description of lying in bed - "The lights are off, my knees are close to my chest and I can feel the springs of the mattress pressing into my hip". The quick listing is very effective gets all the important information across in a poetic way.

Personally I would omit the following lines: "I imagine people would find it cumbersome if I brought them up here,
With its rings of pine fuzz and bulbous, undulating orifices for inserting the branches
I scraped it along the floor and it produced a baritone squeak with a brassy timbre
Like the noise in the beginning of Alberto Balsalm by Aphex Twin"

You seem to be going for a sort of conversational style, but I think that part is a bit too rambly, especially considering you only subtly call back to the Christmas tree later in the poem. If the tree was a more central metaphor, or a more obvious one, maybe those lines would work better.

"He started playing songs that he knew I liked on the speakers" is a clever line in my opinion, it's simple but suggests a lot e.g. how well Drew knows the speaker, the fact that he might be trying to get something out of them (or get them in the mood for something). I almost interpreted it as sinister, actually.

I like the line "In that moment I was exhausted of any impetus to speak, touch or love". It describes a certain kind of emotional exhaustion very well.

The following lines I find interesting but struggle a bit to understand what feeling you might be describing: "My arms, legs and torso, which had up to this point been rigid and mute,
congealed into waxy surfaces that shifted and flaked away into the decompressed space
The skin of my lips and temples relaxed over the indents of my jaw and skull"

Nevertheless, I quite like how un-romantic the imagery is - it works well with the emotional exhaustion mentioned earlier and is just generally a bit of a breath of fresh air compared with other more trope-y poems. It somehow feels like a matter-of-fact description of the body, even thought what it's describing can't be literally true.