Linda Bloodworth-Thomason is apparently fond of playing sappy music during emotional scenes. This happens frequently on Designing Women and to a lesser extent on Hearts Afire.
Scrubs featured a particular few bars of piano music so often in emotional moments that they eventually decided to hang a lampshade on this by having JD say that he always imagines this piece of music playing at emotional moments in his life. From that point on in the series, the music is mainly used in subversions of emotional moments.
That music was featured so much that the actors themselves imagined that tune during emotional moments of their own lives.
Occasionally used during the first five or so seasons, usually during a particularly dramatic Ross/Rachel moment.
There was plenty of this in the episode where Ross's grandmother died.
Same problem when Mr. Heckles dies in season 2 and the group leaves his apartment for the final time.
Usually inverted by The Fresh Prince of Bel-Air: Emotional moments have no music at all, to contrast with the lighter moments of the show, which are usually accompanied with a light hip-hop beat.
This seems to be a staple of Modern Family, particular accompanying the ending voiceovers.
In a non-sitcom example, Minute to Win It does this 2-3 times per episode when everything comes to a screeching halt so the contestant(s) can tell some kind of a sob story about their loved ones.
Boy Meets World was another TGIF show to use this. It was mostly used as transition music at the end of serious scenes though it was sometimes played during the scene if the scene was particularly emotional.
Parodied in the That's My Bush! episode "Trapped In A Closet." Every moment where two trapped characters work out their issues/differences is humorously scored to the exact same sappy music.
Beautifully averted by Frasier which never used music to support scenes. Parodied in one episode where he plays pre-recorded sappy music when apologizing to Martin's girlfriend Elaine on the radio.
Good Luck Charlie slows down the melody of the opening theme of the show at the end of the scene when Teddy finds out that her boyfriend, Spencer, is cheating on her. Most of the time, the theme music is played in a more up-tempo, cheery manner. This one of (if not the only) time the show uses this trope.
Air Crash Investigation, a.k.a Mayday, Air Emergency and Air Disasters, a documentary television show about aircraft disasters usually plays this trope straight at emotional scenes (victim funerals, photos of the plane wreckages, the last few seconds when there's no hope, the first few seconds after the crash, etc).
In The Brady Bunch, many episodes have a mellow or dramatic cue played as Mike or Carol are offering that episode's moral.
Parodied in the Family Guy episode "Holy Crap" in which Peter has a band on hand to play the Sentimental Music at the appropriate moment.
Also parodied in the South Park episode "Cripple Fight", when Big Gay Al is at home moping over his firing from the Mountain Scouts. We hear sappy Full House music in the background, but Big Gay Al soon reveals that it's actually a pianist at his home playing mood appropriate music. He then asks the pianist to play something a little more upbeat.
In Doug, the same music would play during every emotional moment. In this case, it depended on the writing/performances in a given scene. Some of them averted this by being written/performed in a savvy way (usually with dry humor - a great example of them doing this brilliantly is in "Doug's Doodle").
Believe it or not, The Yogi Bear cartoons of the early-60's (along with the late-1980's "New" Yogi Bear Show revival) used this a lot, even in scenes where it made almost no contextual sense (such as when Yogi and Boo Boo are discussing Ranger Smith's birthday in the episode Slap Happy Birthday).