You don't say what meaning you are trying to convey. Both answers could be correct, but probably neither are. "Since" has more than one meaning. It can mean "from a time in the past" but it can also mean "because" He has drunk only rainwater since he is on the island. (This could be correct if using"since" to mean "because". But because of the ambiguity here, it would be better to use "because") He has drunk only rainwater since he was on the island. (If "since" means "from a time in the past" the use of "was" here would suggest that being on the island was in the past, and that he is still drinking only rainwater since that time, even though he is now off the island. So, if he is still on the island, the correct way to say it is (as Nicole states) "He has drunk only rainwater since he has been on the island". However, in spoken English you very well could hear someone use "was" because the context could remove the ambiguity. But definitely not as good. More natural would be something like "He has been drinking only rainwater since arriving on the island." (describing what he is doing on the island) but you could hear Did he drink the beer? No, he has drunk only rainwater since arriving on the island. (answering about his habits in the past)

He has drunk only rainwater since he WAS or IS on the island. He has drunk only rainwater since he WAS on the island. The problem is in the meaning you want to convey. Was he drinking rainwater because there wasn't any other water on the island suitable for drinking? Was he drinking rainwater from the time he arrived on the island? Both meanings are possible with the sentence you provided. Because he could find no suitable water to drink, he has drunk only rainwater on the island. He has drunk only rainwater from the time he arrived on the island.

since being on the island- I would use the gerund in this sentence.

Sentence 1, "He has drunk only rain water since he was on the island." is correct. Sentence 2, "He has drunk only rainwater since he is on the island." is not correct; it must be "He has drunk only rainwater since he has been on the island." These two sentences have different meanings. In sentence 1, the verb "was" is the simple past tense. The state of being on the island ended, and from the moment he stopped being on the island until now he has drunk only rainwater. In sentence 2, the verb "has been" is the present perfect tense. The state of being on the island started, and it has not ended yet; he is on the island now. Starting from the time that he started being on the island he has drunk only rainwater.

English is so complicated! He has drunk only rainwater since he HAS BEEN on the island. WAS would mean that his time on the island is over, it already happened. IS means it is happening right now. But he is still on the island, still drinking rainwater. (Present perfect tense.) He has been on the island, where he has drunk rainwater. He WAS stranded when his boat sank (past tense,) now he IS hoping someone will rescue him (present tense,) he has been on the island for many day (present perfect.)