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Tear Jerker: A New Hope
The Binary Sunset scene. You know the one. It probably has something to do with the music, or the sense of longing to do great things and being tied down by duty. That, people, is why a thirty-year old movie is still making us cry.
A deleted scene where Luke meets his friend Biggs before Biggs goes off to join the rebellion. According to Wookieepedia, it was cut because the other friends-of-Luke scenes were cut. Those were cut for pacing and because Lucas was teased about making "American Graffitiin space". It's still a good scene.
Luke returns home and sees the smoldering skeletons of his Uncle Owen and Aunt Beru after they were attacked by Stormtroopers.
How about Biggs' death? This guy is pretty much Luke's best friend since childhood, yet he's shot down while making sure Luke makes it to the Death Star. Seeing Luke having to swallow his grief for the sake of the Rebellion is just heart rending.
Garven Dreis (Red Leader)'s death is also quite the Tear Jerker, especially seeing as how he knew he was finished and urged on all remaining pilots to let him draw Vader's attention for the sake of the Rebellion.
Obi-Wan letting himself get cut down by Vader but not before giving a smile to Luke who was about to escape. Yeah he became one with the Force and returned later as a ghost, but audiences watching the movie for the first time in 1977 didn't know that.
Interestingly, that scene (and in fact, Obi-Wan's death happening at all) arose from the fact that Sir Alec Guinness absolutely hated the script for Star Wars, to the point where he wanted his part to be smaller and shorter. He convinced Lucas to make that scene end in the character's death so that he'd have very little to do with the rest of the film, and any potential sequels that might have resulted from it as Lucas had already been speculating the possibility of sequels, which is also interestingly why the character Darth Vader wasn't killed in the first film (which was the original idea in the script—Darth Vader, being killed). Rather odd and hilariously ironic inversion going on there, eh?