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Tear Jerker: Mario Kart
Several fans were sad when the Nintendo Wi-Fi Connection was terminated, resulting in no more online play with DS and Wii.
On a happier note, veteran players will tear up when playing Rainbow Road 64 in MK8. Hard to believe it's been eighteen years...
May also apply to Rainbow Road SNES in MK7, hearing old Rainbow Road riffs in newer/returning versions of the track, or even just revisiting other old favorites returning as retro tracks in the later games.
Losing a race, particularly in Grand Prix. This is especially saddening depending on how your character reacts to losing. Rosalina in Mario Kart 8 looks on the verge of tears herself.