The funniest moment in the entire movie may be the "hypnosis" scene as a whole, when Dracula, standing outside of the house, tries to guide Mina out of her bedroom. ("Walk to the door. ... Mina, you are in the closet.")
Dracula: You will be my bride throughout eternity. We'll share the endless passion of immortal love.
Essie (Mina's less attractive maid): Oh I can't wait!
Dracula:[stares at her in surprise] Not YOU!
Takes her back inside and throws her on the floor, and carries Mina out, walking very fast
Van Helsing telling Seward and Jonathan "She will become one herseeeelf!!!" in an overly dramatic and hammy fashion. When asked to repeat himself, he says it in the exact same way.
The whole exchange between Jonathan and Van Helsing when Jonathan has to kill Lucy. Van Helsing tells Jonathan that he (Jonathan) must be the one to do it, and sets Jonathan up in front of the body with a stake and mallet. Van Helsing then goes to stand behind a pillar, and Jonathan bangs the stake into her heart and a veritable geyser of blood erupts, drenching him.
Jonathan: [bangs the stake again and is further drenched]
Van Helsing: She's almost dead!
Jonathan: She's dead enough.
The scene was also, it should be noted, Enforced Method Acting. Steven Weber had no idea how much blood was going to be shooting out, and you can tell he's struggling to keep a straight face.
And then shortly after that...
Van Helsing: I have been to many, many stakings. You have to know where to stand. You know, everything in life is location, location, location...
"Renfield, you asshole!"
After Dracula breaks up his brides's attempts to "wrong" Renfield's brains out, they begin to glide out, with the same ethereal creepy music that played when they came in. After a second, Dracula snaps "And stop that!", and they just trudge out.
The Running Gag with Dracula and Van Helsing each trying to get in the last word in "ancient Moldavian." Also possibly relevant to the plot, as Van Helsing realizes that Dracula, like the mysterious vampire, never gives up.
Dracula's attempt to hypnotize the woman at the opera.
Dracula:...And you will remember nothing of what I have told you.
Beat. The woman nods, and walks over to Dracula as though seeing him for the first time.
Woman: May I help you, sir?
Dracula: [mockingly] May I help you? Why did you not tell him?
Woman: About what?
Dracula: About... the message.
Woman: For who?
Dracula: Never mind! I will tell him myself. And for your miserable performance, you shall receive no tip!
Woman: No tip!?
Dracula: Oh, now THAT, you remember!
To compound the comedy, just after Dracula introduces himself and begins a conversation with Dr. Seward, the woman - back in the trance - returns and delivers the message after all.