"My chair is gone" is correct - to say it with other words: It is not here, it has disappeared, it has vanished. A clause in passive speech, present tense. In the current situation it "is" not there anymore, it "is" gone.
"My chair has gone" is used as well but it is an idiomatic expression. As you know a chair cannot "go" by itself, it has been taken away by someone, i
The more I'm thinking about this the more I guess that "has gone" might be a confusion which arose out of speech contexts: It is gone = It's gone This is the passive construction - the abbrevated "It's" though can also represent the form "It has" and because of the fact a past participle follows, that might have led to a confusion with Present Perfect active speech.
If you say "has gone" you will seem to be saying that the chair decided to get up and leave all by itself. In this universe that is impossible as far as I know. : )