Sims with no sense of humor tell horrible jokes so they tend not to tell them. They also don't enjoy the jokes of others. Humor is simply wasted on them.
Sims have interactions like "Ramble Endlessly" and "Share Trivia", which bore ordinary Sims but delight others with no sense of humor.
Sims have "Bore To Death" interaction that make other Sims pass out for a moment afterwards, which can make celebrities to be publicly disgraced when they do[TS3:LN].
Sim will negatively react to a humorous social interaction unless the relationship with that Sim is high enough.
Sims with no sense of humor can "Have a Blast" for 15 hours instead of 8. This is because Sims with this trait find the simplest things interesting and exciting.
Kids with no sense of humor cannot go for a joy ride, preside over royal court, or other child games[TS3:G].