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.
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.
Rhea Perlman as Zena, Louis' girlfriend (and DeVito's Real Life partner.) When Perlman starred on Cheers, DeVito was going to play Nick Tortelli, but then he became a hot movie property.
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 exent of Soap, which by competing with The Facts of Life it hit rock bottom and was swiftly canned).
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.
Robin Williams was supposed to play Bobby, but he had to decline when Mork and Mindygot picked up.