Adored by the Network: While being one of the "bottom 10" shows of the 1981-1982 television season, somehow NBC saw a worthy product on it and Un-Cancelled the show. Though it didn't fare better with viewers and NBC was forced to cancel it (neither Cheers nor Family Ties got better numbers, but were spared by their numerous Emmy nominations and both ran much longer than Taxi).
Nick @ Nite and later TV Land kept Taxi on reruns for around a decade, building a large new fandom for the show. It helped that the show had a rather shiny picture of NY, making it look like it wasn't made in The Seventies.
Channel Hop: From ABC to NBC in its last season. Ironically, the show which beat it in the ratings during the third season on NBC, Diff'rent Strokes, moved to ABC during its eighth and final season.
Doing It for the Art: While quite commonplace nowadays, when JCW Productions pitched the show to HBO in 1982 (after being cancelled by ABC), no cable network had aired a sitcom (mostly because almost no one had cable back then) thus being considered a very daring (and risky) move at the time. However, Paramount decided to have Taxi on either CBS or NBC, the latter of both ended up running the show for a year.
Enforced Method Acting: One episode called for Louie DePalma to whisper something to Elaine Nardo, and for her to respond by slapping him and saying "That's disgusting!". Danny DeVito whispered such sweet things to Marilu Henner during rehearsals that she was genuinely shocked when he started whispering not-so-nice things, and she blew several takes because of it (to DeVito's delight).
When Carol Kane joined the cast as Latka's girlfriend Simka, Andy Kaufman taught her their country's "language" by inviting her to dinner and refusing to speak English or let her do so.
In one episode, guest star and real-life boxer Carlos Palomino accidentally delivered a real left hook to Tony's face. You can see Palomino pull his hand towards his mouth in horror for a second on realizing what he did before getting back in character and turning around to exit.
Real-Life Relative: Louie's girlfriend Zena was potrayed by Danny DeVito's real-life wife Rhea Perlman.
Also, DeVito's own mother Julia appeared as Louie's mom in a few episodes.
Tony Danza's son Marc Anthony appeared in two episodes of the second season, appearing in one as the kid Tony wanted to adopt.
Averted with Christopher and writer David Lloyd, with both being unrelated. Ironically, David is the father of another (also unrelated) Christopher [Series/[Modern Family Lloyd]].
Screwed by the Network: An unintentional example. To boost the show's ratings, ABC placed Taxi for its' third season on Wednesday, opposite NBC's hit show, Diff'rent Strokes. To be short, ratings dipped considerably (but not to the extent of Soap, which by competing with The Facts of Life it hit rock bottom and was swiftly canned).
Troubled Production: Andy Kaufman never took the show seriously and often made ludicrous demands, most notably his showing up to the set in-character as Tony Clifton and causing utter chaos. To a lesser extent, Jeff Conaway suffered from a heroin addiction that kept leaving him unable to work for long periods, until he was fired.
Un-Cancelled: The show was abruptly cancelled by ABC after its' fourth season, but it was picked by NBC for another year before being cancelled for good.
What Could Have Been: The Season 1 episode "A Full House for Christmas" was supposed to guest star TonyClifton as Louie's brother, but Clifton's behavior on the set was so awful that he was fired; he wound up being escorted off of the Paramount lot.
Originally "Touchdown" by Bob James was going to be the title theme but the producers liked James' slower music for the episode "Angela" and chose it instead.