The sequence in the special Garfield's 9 Lives involving Diana's Piano. If the music doesn't get you, the added visual at the end probably will.
There is a sequence in the television special Here Comes Garfield that qualifies as well. Before Odie is dragged away to certain death by the pound employees, Garfield lays next to him to comfort him. Garfield then cries for him, because there was nothing else he could do. Also, Garfield's final lines before Odie is taken away really tug at the heartstrings.
Garfield: I'll see you, Odie. Sometime, somewhere... so long, old friend.
The final scene of Garfield on the Town (based on the aforementioned storyline of the strip) in which Garfield awakes the next day afraid that his encounter with his mother the previous day could have been a dream, then notices she was watching him thought the window of the house and quickly retires. Cue sad music, Garfield goes to the window to watch his mom leaving. Although it may have been the case of the Latin American dub because Garfield last words instead of "Thanks mom, for everything" were more along the lines of "Thank mom. Farewell."
The Christmas special? Garfield - who we've always known as a selfish, greedy, gluttonous cat - gives Grandma Arbuckle a bundle of long-lost love letters from her late husband; she calls them "the nicest presents I could have received". And then Odie - who's always been taking the brunt of Garfield's abuse - give Garfield a thoughtful gift as well.
Grandma:[reading] "My darling; if the sea were of ink and the sky of parchment, I could not begin to write my love for you..."
Also from the Christmas special is Grandma reminiscing about her deceased husband to Garfield, and confessing that Christmas was his favorite time of the year... and the time of year she misses him most. *sniff*