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.