To preserve the film's PG-13 rating.
To show the injury would have been too gory for audiences, especially on top of Katniss' burns. Apparently a more serious version of his leg injury made it to production but ended up on the cutting-room floor, according to the film's senior make-up artist Ve Neill. It's also noteworthy that losing the leg entirely was apparently never discussed as an option.
On how much gore she knew she could get away with in regards to the blood and wounds of the movie:
I’m very aware of it because I had just done The Amazing Spider-Man which is also PG-13 and we did a lot of prep work there. I knew the coloring the blood had to be, I knew how much we could do, and what we could do and couldn’t do with the blood. I really knew our limitations ahead of time and it was good that I did because most people don’t know that going into a film if they haven’t actually done it, you know? It winds up getting taken out or they can’t use it so it was good that I had that knowledge going in of what we could see and what we couldn’t see.
On Peeta’s leg injury:
No, we never talked about losing the leg. We did do part of a huge wound but we wound up not seeing a lot of it which was fine because I think it was too much. I think they showed just enough of everything to really get the point across of how severe the injury was, along with [Katniss’] burn. I think all of it worked out really well.
Ve Neill Talks The Hunger Games and More
To remove the need to continually have to explain his leg
In a Q&A Session, the film's producer Jon Kilik spoke+ about the need to not have things affect the subsequent films. He was presumably referring to the fact that if you permanently injure a character at the start of a film series, then you'll have to keep finding ways (and taking valuable screen-time from each subsequent film) to explain it, even if it's no longer plot-relevant.
Q: Why didn’t Peeta lose his leg in the movie?
[I was so distracted by trying to remember how much we knew about Peeta’s leg having been mauled by the muttations and subsequently amputated and replaced with an artificial limb, and when we knew it—that I can only paraphrase Kilik’s response. He basically said that they had made some decisions about that that would affect the subsequent movies.]
A Peek Inside ‘The Hunger Games’ with Movie Producer Jon Kilik
+Unfortunately we don't have the exact quote but one of the attendees did write down their recollection of what he said