It's a comedy show, but that doesn't mean they don't take facts seriously. As I said in another comment further down, QI is:
a show with a team of good researchers, they can on occasion be wrong (and have corrected themselves for mistakes in previous shows in the past), but they won't put a fact into a show without getting good sources for it
So while they may be wrong in this instance, the fact that they are a comedy show isn't reason enough to think so.
Anyway, it's not true. Air in the cabin is bled off the engines, from the early stage of the compressor, then pressurized and heated. It's not one cabinful of air that was in it from the ground endlessly recirculated.
QI also told the world that the Internet uses GPS to route packets (tho' some devices may use GPS for its clock signal, packets don't know geographically where they are).