Technically, unless I'm very much mistaken, starvation is the body eating itself for nutrients. Much like dehydration. To determine the answer to this question, we have to confront the precise nature of his healing factor, and the facts thereof: It shouldn't work. Wolverine isn't shown ingesting any additional nutrients that his body could use to regrow lost appendages or other injuries. As such, we have to assume one of two possibilities: he gains that mass from an alternative dimension or some such other energy source in order to fuel that regeneration. Or: His mutation has grown an organ within him that produces it's own nutrients, making his own food when necessary. The latter is far more likely, so yes, more likely than not, Wolverine does not necessarily NEED to eat.
However, you also have to factor in that he's constantly suffering from adamantium poisoning due to his skeleton coating. Therefore, while it may heal severe injuries, it's possible that it would not heal starvation or dehydration while he is in possession of said skeleton til the latter stages where his body begins to shut down. So he may end up in mindless agony til he eats.
I hope that was a sufficient answer, but keep in mind that's totally speculation.
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