Aww...
I'm probably "built close to water" as we say in Germany. I'm easily triggered by emotional stories, be it books, movies, movies, music and even games.
For games the most infamous examples would be:
FF7 - who's played it will probably know when and why.
TWD - mostly Season 1 on several occasions, but also the later parts.
PS:T - mostly the tragic in the both the impossible love stories, but also when the background of the relationship between Dak'kon and the Nameless One was revealed (the Unbroken Circle..).
The Witcher (3 was worst, but all games had some moments) - after living and suffering with the characters through thousands of pages of Sapkowsiki's brilliant writing and having this spirit perfectly captured by the games.. well...
There are more, but those are probably the more well-known titles.
That said, tears aren't the only expression of empathy that games can trigger. There are quite a few more scenes in games that left my mouth dry and my heart pounding.
There is, for instance, one scene in The Cat Lady. The protagonist comes home in a very dark mood and just wants to calm down and relax. As the player you have to help her. Find the things that help her find peace, avoid things that upset and depress her (the appartment is not... in the best state). Then someone starts pounding at the door...
This scene is hands down the best depiction of what depression is like and what people suffer through to deal with it I've ever experienced in a game. And I speak from my own experience. It's so pragmatic and ... lacking for a better word: "the thing", it scared me and at the same time gave me comfort because I felt so understood. No tears here, but a very great achievement for a game.