In Vandread, Kahlua a.k.a. Pyoro-2 was born in an elevator. The male lead is initially tasked with handling the delivery, but he manages to hand it over to his love-interest fairly quickly.
Fairly narrowly averted in the very beginning of Kyo Kara Maoh. The scene (a warm conversation about baby names in the back of a taxi) is given without a shred of context; we work out who the principle players were in subsequent episodes, and see the scene again, de-blurred and with a lot more context, a long while later in a flashback episode.
Averted but referenced in Die Hard. After Hans and his crew have taken the Nakatomi office workers hostage, Holly McClane Genaro tells him one of them is pregnant. Hans immediately rolls his eyes, expecting this to take place before Holly tells him she's not due for a couple of weeks, but that she need something more comfortable than the floor to sit at. Hans agrees, and has his mooks move a couch from the lobby to the main hall.
The going into labor in a taxi version is used in Look Who's Talking. (The taxi driver sticks around and the mother, who has broken things off with the baby's father, eventually falls in love with him.)
Fran went into labor in a broken elevator in The Nanny, but they got her out in time for her to deliver in a hospital.
Malcolm in the Middle: Lampshaded by Hal at one point. He saves a pigeon caught in plastic packaging at the side of the road, and says, "Well, I wanted to deliver a baby in the back of a taxi, but..."
This happens in The Suite Life of Zack and Cody, where the baby is born in The Tipton's elevator to a mother called Mary and a father called Joseph at Christmas. The theme was broken by the baby being a girl.
Happened in Early Edition. The power goes off, and Chuck and some pregnant lady are trapped in an elevator. Chuck, unfortunately, has a weak stomach and passes out. Gary goes down the elevator shaft, delivers the baby, and leaves, leaving Chuck with the lady and her baby when the elevator doors finally open after everything gets working, making Chuck look like the hero.
Night Court. When Public Defender Christine Sullivan is pregnant but insists on working regardless, the others plan "Operation Stork" — a ridiculous Crazy-Prepared scenario should she suddenly go into labor. Unfortunately they forgot to include the elevator in their preparations.
An All in the Family episode has Archie getting stuck in an elevator with several people, including — yes — a pregnant woman who goes into labor.
Gloria manages to get stuck in a phone booth after calling the doctor to say she's in labor. They get her out to go to the hospital though.
Alluded to in Stargate Atlantis when a misfiring security system put the entire city in lockdown, locking in Sheppard and visibly pregnant Teyla alone in the lab.
Sheppard:(glances nervously at Teyla) Teyla: What? Sheppard: Sorry, it's just that when something like this happens in the movies, the pregnant lady goes into labor.
The 'giving birth in a taxi' variation happens in an episode of Frasier. However, in something of a variation it's the taxi driver's baby, not a passengers. Also, the two highly educated persons who added a doctor in front of their names were useless, while the retired police officer helped her through the crisis.
In My Name Is Earl, Billie goes into labor on the hospital's broken elevator (during Earl's coma dream). Earl has to help her...and he is proudly holding a healthy baby boy afterwards in the dream.
Home Improvement had a variation on this. Heidi goes into labor while she, Tim and Jill are driving through the middle of nowhere to go to an awards dinner and she gives birth in a roadside gas station. Tim becomes the Delivery Guy.
Averted on Kung Fu: The Legend Continues when an elevator is stuck between floors with a visibly pregnant woman, a jerk, and Kermit Griffin, a tough guy. The woman gets upset, the jerk is rude to her, and Griffin threatens him for being rude. All are rescued, and the woman does not go into labor.
Joan of Arcadia features a quite cruel variation: after the baby is born, the mother refuses to look at it and demands it be taken away.
Punky Brewster: in the episode "Ouch," in which Punky breaks her leg and Cherie feels guilty about it, at the hospital Cherie is in a stalled elevator with an expectant mother who is going into labor. Cherie helps with the delivery.
A M*A*S*H episode has Hawkeye helping a pregnant Korean woman deliver (with no help from a squeamish Radar) while aboard a bus en route from her village to the 4077th.
This trope is alluded to in Elite Beat Agents where a taxi driver is asked to take an expecting woman to the hospital. (However, regardless of whether he succeeds or not, she does not give birth in the taxi.)
King of the Hill: Bobby amuses himself by pressing the buttons for every floor inside an elevator, then gets bored and leaves. A few seconds later, a man and a doctor wheel a birthing woman into the elevator. The door closes. Ding... Ding... "WAAAH!"
In 2013, a woman gave birth on a packed (until they emptied it at the nearest station, anyway) commuter train at rush hour. The train did sit at the station for an hour, though suggesting that this was only a 'sudden' birth relative to the norm.