I thought of this earlier while I was watching "Random Harvest" (thanks pgoat!) It's really hard for me to narrow it down to just one, because I love a lot of the sappy old movies (think Frank "Capricorn") but I also like Metropolis, All Quiet On The Western Front, Citizen Kane, Battleship Potemkin, etc. I think overall, my favorite old movie must be City Lights. I think I could watch it every day and the final scene would still cause a lump in my throat.
So what's your favorite? And by "old movie", I'm generally thinking no newer than early 1950s, but this isn't a rule at all. If you consider Encino Man old, that's perfectly fine too. I own a decent set of old movies, but I'm always open to see others.