There are two independent objections.
• They're not as "clean"
This first objection has to do with elegance, understanding, and pedagogy.
Proofs by contradiction sometimes have an extra step. Some can be shortened by making them direct. That's not always the case, though. Occasionally a proof by contradiction is shorter.
Even when a proof by contradiction is necessary, it may be hard to follow. The whole proof is hypothetical. It discusses things that turn out not to be possible. Take Euclid's Elements, Book III, Proposition 10 for example.
A circle does not cut a circle at more than two points.
A direct proof may be possible when you've got coordinates for your points, but Euclid worked 1900 years before they were invented, so he gave a proof by contradiction. He assumes that two circles to meet at three points B, G, and H. Because it's a hypothetical situation, the diagram he draws is impossible. Yet he perseveres and eventually reaches a contradiction to the previous Proposition 5 "...therefore the two circles ABC and DEF which cut one another have the same center P, which is impossible."
It's easier to follow a direct proof, and that means it's easier for students to understand it.
• Nonconstructive existence proofs
One way to prove something exists is with a proof by contradiction. You assume it doesn't exist and draw a contradiction, and therefore you conclude it does exist. Knowing something has to exist may be helpful is some way, but you can't do anything with it if you don't know how to find it.
Euclid had none of those proofs. The Elements consists of constructive proofs.
The most famous nonconstructive proof is Brouwer's fixed point theorem. Brouwer showed that under certain situations, a function had to have a fixed point, but his proof was nonconstructive. He was not satisfied with his proof and endeavored to restrict mathematics to exclude these nonconstructive proofs, thereby founding intuitionistic mathematics.
• Other objections?
There may be other objections, especially those of the first category that claim proofs by contradiction aren't as nice as direct proofs. Brouwer's objection is more serious. Even if you don't go as far as Brouwer did and completely reject nonconstructive proofs, you'd probably still prefer the constructive ones when available.
