Does having proficiency in another language help officer promotion?

Completely agree with COL Vincent Stoneking. For the promotion boards I've been on I can honestly say that language proficiency has never come up once.

I was involved once in a selection board for enlisted linguists where language proficiency was considered, but that was a special case and it wasn't for promotion.

Your Mandarin might get you an edge in an assignment, but not promotion.

It won't matter in a "fully qualified" board - unless in some future timeline, being multi-lingual becomes a promotion requirement.

In a "best qualified" board - Honestly, it depends and you'll likely never know... But it will come down to the instructions given the specific board, the competitiveness of the cohort, and the members of the board's views. IF I were sitting on a general purpose board, I would -personally- give it NO weight whatsoever, unless the board instructions told me to do so, and even then I'd likely downplay it. If it were a FAO board, or one limited to MI branch, I might well give it great weight in a competitive cohort. I could see it being of weight for various special-purpose boards.

That said, I made a point of getting a DLPT rating for French on my ORB, because you never can tell. I have no idea if it made a difference. I expect that it made none whatsoever.

Some languages open up the ability for foreign schools. I know there are multiple programs to get ILE through foreign schools if you are proficient in French.