What does "to be off with someone" mean in this sentence "Ever since I said no to him, he's been very off with me."?
33. slightly abnormal
34. not up to standard; not so good or satisfactory as usual; inferior or subnormal
It's between these two definitions. If I say, "He's been off with me," or, "He's been very off with me," it means that things are not the same between us, that he hasn't been treating me like he normally does, that there's an awkwardness, issue, or something going on with him that's affecting his behavior towards me, like maybe he's upset with me about something I said or did, which I maybe don't even know what that something is because I didn't realize he took offense, got hurt feelings, or had whatever emotional reaction he had that's making things off between us as he maybe hasn't told me, or maybe he's just been in a funk or in a funny mood lately about something completely unrelated to me, but whatever it is up with him, the way he's been with me, the way our relationship has been lately, has been not like it usually is, and not in a good way. It's that strangeness in him and how it's affecting how he's treating me by his treatment of me being substandard that I'm commenting on.
Being that the usage is somewhat slangy or informal, I believe this definition from Urban Dictionary sums it up, as well:
To be weird, off center, not normal.
"she's a bit off"
7. Inappropriate; untoward
I felt that his comments were a bit off.
As used in your example, the meaning is not quite as above, but similar. It sounds like the person is being unfriendly, distant, and awkward, rather than inappropriate.
As a native speaker, one aspect I'd often understood of phrases like "a bit off" was by analogy to something like milk or cheese (or indeed other perishable items like meat) that has turned sour or otherwise started to acquire an unpleasant flavour or odour, which could also be described as "a bit off".
Conversely, when it has curdled and become so sour nobody would have any doubts it had "gone off", it would be considered "very off".
By analogy, when someone is interacting with you in a different (and less pleasant) way than you would normally expect, you could describe that interaction as "a bit off" or even "very off" depending on the circumstances - not the interaction you were expecting, and either subtly or very noticeably different to the normal friendly response you expected.
In essence, it is suggesting that the relationship/friendship/interaction has at least started to turn sour.
I think the cited usage is contracted from He's been very offhand with me lately...
offhand - Ungraciously or offensively nonchalant or cool in manner.
I'd say the contracted version is slightly "slangy". Also it's probably best avoided by non-native speakers (in favour of offhand) because in certain contexts the intended meaning might not be clear (since off has an awful lot of different meanings! :)