Obviously not deliberate, but since Peter Capaldi was cast as the Twelfth Doctor, it's become extremely retroactively funny how often Malcolm's rants involve time travel and other science fiction tropes.
Even more coincidentally, in one of the DVD commentaries, the cast jokes about Capaldi someday being cast as the next Doctor.
Enforced Method Acting: For the enquiry episode, the cast were given no rehearsal or preparation before filming. Furthermore, there were no opportunities for the actors to talk to one another between re-shoots, and were kept separate from the "witnesses", like an actual enquiry.
Foreign Remake: In 2006, ABC ordered a pilot for an American version of The Thick of It starring Oliver Platt. By most accounts it was terrible and consequently was never picked up. Veep fared much better, though it's more of a Spiritual Successor than direct remake.
Role Ending Misdemeanor: In 2005 Chris Langham was charged with downloading child pornography. Rather than make a full second series without him, the producers made two hour-long specials for which they put him on a bus, leaving the door open for him to return if he was found not guilty. He was found guilty at his 2007 trial and served 15 weeks in prison, and his character Hugh Abbott disappeared from the show never to return.
Shout-Out / Take That: The season 3 episode in which Nicola and Peter are interviewed by Richard Bacon contains references to two other Five Live broadcasters, Simon Mayo and Mark Kermode. The reference to Kermode is only an off-hand, blink-and-you'll-miss-it comment about his supposedly "massive" hands by Ollie, but it seems to have taken on a memetic life of its own around Kermode.
The show takes fairly regular potshots at Top Gear, especially Phil being disparagingly compared to James May and Malcolm saying Richard Hammond needs a punch in the face. In another episode, Malcolm says that he hates Steve Fleming "as much as James May presumably hates himself."
Jamie, after Ollie spectacularly fails to find out opposition secrets by wooing Emma, tells him: "You must be the shittest James Bond ever! You're David fucking Niven!"