Is it acceptable in English to say "I am off" in meaning that I'm tired or something like that? I heard it from non native English speaker, that's why I'm not sure about it.
Well, I'm off!
Means that you're leaving. See: TheFreeDictionary - I'm off.
This song sounds/feels a little off.
Means that there is something not quite right about this song to you. It feels weird. A thread on ELU about that: What does the idiom “to sound a little off” mean?
If you wanna say you're not feeling well, you could say:
I'm feeling a bit under the weather, I think I'm going to go home.
It's possible you may be thinking of
I'm a bit off (BrE)
I am feeling unwell
and therefore the speaker is feeling tired as a result.
To only express tired might be
I'm tired (obviously)
I'm knackered (BrE)
Usually expressing tiredness and sickness are kept separate.
As pointed out, simply using "I'm off" would usually mean
I'm off (to some destination)
Yes, I think so. It may be a little unusual, but I think it would be understood easily by a high percentage of native speakers.
The more "usual" statement would be "I feel [a little] off" or "I'm a little off" or "I'm off my game", etc. but I think "I am off" or "I'm off" are adequate and perfectly understandable.