About 1 hour and 44 minutes into the movie, we see Lord Kelvin in his office playing with the metal spring which we know as the Slinky. As he's playing with it he's humming the song "It's Slinky!" from the old Slinky television commercial.
When Fogg is stuck in San Francisco as a beggar, you can see on the wall behind him a poster advertising the mineral water "Hunyadi Janos". Named after a 15th century Hungarian general, this water had been discovered in 1862 and introduced to the US around the 1880's, whereafter the Hunyadi water became a widely used laxative for the next couple of decades. In fact, you can also note the word "laxative" on the poster.
The trivia items below may give away important plot points.
The French balloon man who calls the main characters insane as they jump into the basket is a cameo appearance by Sir Richard Branson. As well as being the boss of Virgin, he is also famed for his sometimes life-threatening ballooning record attempts.
John Cleese is the fifth Monty Python member to appear in an adaptation of Around the World in 80 Days. Michael Palin starred in a BBC documentary in 1989 where he attempted to travel the same route as Jules Verne's Phileas Fogg. The doc also featured appearances by Terry Jones and Terry Gilliam. Eric Idle starred in a 1989 mini-series of Around the World in 80 Days as Passepartout.