Why did Homer, Moe and Wally cover their mouths when talking about the Super Bowl?

The episode was aired on the same day as Super Bowl XXXIII and the writers want the episode to seem current. They had already animated the scene and it would have been very awkward to re-animate the whole scene just to put the team names in.

So they just held up a glass over the mouth so they wouldn't have to animate their mouths to match the sound.

They did the same when Wally was mentioning the Clintons as Bill was being impeached at the time so they weren't sure if he would still be President when the program aired. They were also poking fun at the fact that Hilary mightn't be his wife either when it aired.

A typical Simpsons episode "takes 6-8 months to do one episode" according to Simpsons creator Matt Groeing. Of course, they are creating multiple episodes at a time, but from the starting ideas to finish episode, it takes months.

The average time between the last game of the NFL Conference Championship and the Super Bowl is 2 weeks or less. In 1999, the Conference Championship was January 23rd, and the Super Bowl was 7 days later, on January 30th.

This was the day before the Simpsons episode "Sunday, Cruddy Sunday" aired on January 31st.

The reason they cover their mouths, out-of-universe, is simple. They wanted to air an episode about the then current Super Bowl, without the complications of redrawing the scenes in a week, or multiple times. By hiding the mouths, they could quickly dub the team names in. Alternatively, they could have animated the mouths to fit multiple team names, but that would be much costlier.

According to Wikipedia, who cite the episode writer Mike Scully's Commentary for "Sunday, Cruddy Sunday", in The Simpsons: The Complete Tenth Season DVD set:

Originally, the characters would be saying something else, however, because the staff wanted the episode to be "current", new dialogue was recorded for the scene. Because there was no time to animate the scene from scratch, the staff simply made the characters hold a glass in front of their mouths while saying their lines.

It's comedy. It's a joke. Contrary to the other two answers, this was not any sort of scheduling or technical limitation.

The joke is that the audience knows cartoons such as the Simpsons are created months ahead (South Park being one of the few exceptions to this rule).

The mugs-in-front-of-the-mouth is just a very obvious nod to that. Note the comedic timing of the inserted audio for each statement of the team. It's simply a joke.

In fact, it's a more complicated solution than other solutions:

• cut away from the character as they state the team
• animate the mouth generically
• don't use a close-up as they speak the name
• don't repeat the gag so many times (Of course, that's what made it a joke in the first place).

As for Clinton/Hilary part...the joke there is that the character Fred Willard doesn't get the joke the show is making. His character is rather oblivious throughout the entire show. Again, because it's funny. :)