What's causing this mesh error? I've tried a few different things like weight painting but it doesn't seem to do anything helpful unless I'm weight painting it incorrectly.

I was doing a quick rig of this model port when I came across this issue.
There are 3 body meshes that go together and I tried to merge together to see if that would work but it does not.

It's only when you bend the model this issue occurs.

Here's the .blend file and textures in case you want to take a look.
https://www.dropbox.com/sh/fgubdg3k7hz50pq/AACfqDFhF5ojuRpLGqNtu0M7a?dl=0
Your body mesh seems to have a lot of duplicate faces.  You can see this even by moving around the model in the viewport...those weird marks are the geometry fighting:



Another way to see is if you select a face on the torso and choose "Select->Select Similar->Material" and you'll get most, but not all, of the faces.  

I would suggest selecting ALL of the vertices of the body model, then choosing Mesh->Vertices->Remove Doubles.  That will remove the extra geometry.  

Unfortunately then you will have patches of the torso that are a different material (so they'll be purple) but you can select one of those, again do "Select->Select Similar->Material" and set them to the same material as the rest of the Torso.  Or vice versa, depending on what you want. 

You might have to re-rig after removing the doubled geometry, but it will solve your rogue pieces sticking out.