Why does Wolverine drink alcohol?

From his wiki:

Wolverine's natural healing also affords him the virtual immunity to poisons and most drugs, except in massive doses. For example, it is extremely difficult for him to become intoxicated from alcohol.

So to answer your question: Intoxication is a state of being poisoned. Shouldn't his powers prevent him from it?

They do, largely.

But your first question was: why does he drink?

The fact his body makes it hard for him to get drunk doesn't mean there's no reason for him to drink. There's plenty of reasons to drink, including:

1. Enjoying the taste
2. Enjoying the atmosphere of the bars, as opposed to other places
3. Wanting to be drunk, given Jean's death, even if these drunk periods are short-lived
4. Having a "mental" addiction to the alcohol.

Ultimately, just because his body can fight off being drunk quite quickly doesn't mean there's no reason for him to drink at all.

From Wolverine V3 #33

In this scene Mystique seems to be sad that Wolverine abuses pills and drugs. It doesn't seem logical for her to be sad for they can't really hurt him. But maybe the backstory might be suggesting him having some probably romantic problems. But still, this comic shows that Wolverine can certainly drink for reasons other than taste.

In Wolverine #900, Spider Man and Wolverine go to a bar. Wolverine constantly drinks, get drunk, and get back sober right after. This is why he drinks a lot when he does.

Again, in X-Men Animated Series, we see him getting shot with tranquilizers and pass out. So even though his powers give him strong resistance and fast recovery, it doesn't mean he is completely immune to all kinds of toxications.

Long story in short, yeah he can get drunk, even though it is for a short time.

Wolverine's adamantium skeleton is toxic and a good chunk of his healing ability is constantly in use countering that effect. This is explained in more detail in the months preceeding his death as Beast invents and administers an antidote when Wolverine loses his healing factor permanently. Since his healing factor could be overloaded (Re: Post-Adamantium loss in X-Men #25 (1991)) it stands to reason that his adamantium skeleton allows him to enjoy alcohol and feel at least some of it's effects. I assume he drinks beer for the taste.

This same basic question was asked on SE Science Fiction & Fantasy. The accepted answer stated the following:

Yes - it's possible, although very difficult, for Logan to get drunk.

Due to his healing factor, alcohol and other intoxicants do not affect Wolverine like they would a normal human. His system, even at low efficiency rates, burns the substance out of his system faster then he can feel the effects. Amounts that would kill a stout drinker from alcohol poisoning barely give him the "buzz" effect that would constitute "drunkenness".

That said, it IS possible under the right conditions for Logan to get drunk. It would require a continuous supply of the substance, and in a faster rate then what his body can burn out/consume.

Generally my reading (over many decades) is that Wolverine is trying to, and succeeding at, getting drunk -- but is usually shown with the enormous number of bottles it takes for him to succeed at it.