I always thought that part was funny, where David is in the meet and he has a shirt on. Guess NPH didn't feel comfortable bearing his baby fat to the moviegoing audience.
"First you ask if you can be red, knowing that I'm always red."
*spoilers*
He takes his shirt off later in the movie, I think they just had him keep the shirt on to distinguish him from the other swimmers. I don't think NPH has ever really had a problem with his body. In tons of Doogie Howser episodes we see him changing or without a shirt on.
First, he is not at a swim meet...he is at tryouts, which he had displayed so far in the movie that he is not comfortable about his abilities (at least not as comfortable as his dad is). Second, he is still shy and unsure of himself, so he leaves his t-shirt on - something that makes him stick out as someone with not enough confidence. Kind of hiding behind it...
The fact that later on in the movie, he goes to challenge the punk-kid that teases him, he is very confident and in better physical shape and he swims without it.