If you have set of identical riders, and one rider puts a slower tyre on, they will lose if no-one gets a puncture (let's simplify and assume a time trial so we can ignore effects of the peloton). Anti-puncture tyres are slower as seen in R. Chung's answer. Of course for a real race of merely very similar riders who may be having good or bad days, it's not quite as clear cut, but it undoes a lot of the marginal gains that are so popular.
Punctures in races are actually quite rare. The only time an anti-puncture tyre would come into its own is if all the other riders suffer punctures. For a normal road race, that would require sabotage.
Here's an old article that might be of interest.