Near the beginning of Paulie, when Marie has to watch her bird get taken away in a van, and she's crying and her parents have to hold her back and keep her from chasing the van... as a pet owner myself, and knowing that people really have had their pets taken away for unjust reasons (usually to be euthanized), this whole idea is just deeply, deeply disturbing to me. Probably the only reason I can still stand to watch the movie is because Paulie does reunite with Marie at the end.
Also, the end of The Plague Dogs, and that song... when I started up the movie a second time because my brother hadn't seen it yet, I just about broke down as soon as it started because I noticed they were playing that song during the opening credits too.
Ash sacrifices himself to save the Pokemon and everyone (minus Mew and Mewtwo) start crying. Many people have called this the saddest moment in all of Pokemon.
Personally, that scene never really had that kind of effect on me. Jirachi Wishmaker, though, I have a hard time even watching because I just find it so incredibly sad that Max and Jirachi get so close to each other, and then Jirachi has to go to sleep for 1000 years, which means they can never see each other again.
Also, I still think the saddest Pokemon moment is the "Pikachu's Goodbye" song, closely followed by some of the actual release episodes (Charizard, Arbok and Weezing), as well as that one scene in "Address Unown" where they saw Larvitar's memory of being taken from its mother.