The new (and very beautiful) Disney logo will always be associated with Pirates of the Caribbean: Dead Man's Chest to me because it was the first movie to use it.
New Line = LOTR Paramount = Star Trek Universal = Jurassic Park (I love the logo they used in the early/mid 90s a lot more than the current one, which I find too cartoony) Fox = Star Wars
New Line - Lord of the Rings Paramount - Indiana Jones Universal - JAWS Fox - Star Wars RKO - King Kong (The RKO logo is my favorite studio logo of all time)
The new animated flyby paramount logo definitely makes me think Star Trek. The old Paramount makes me think Raiders of the Lost Ark.
The Warner Bros. logo used on all their DVD's (when you first put the disc in, with the piano music) makes me think of Contact, the first DVD I owned and the ONLY one I owned for 2 months, so I watched it a lot.
HBO = The Beastmaster This is from a stand-up comedy joke "Whats HBO stand for? 'Hey, Beastmasters On!'". I doubt HBO had anything to do with Beastmaster theatrically, but thanks to that mnemonic comedy routine, I laugh a little whenever I see the logo.