My problems originated from the fact that I used MSAA on all my render textures, even though I didn't need it in most of them. Now, I have defined every render texture without MSAA, except for the one I render my main scene to. The second problem I had was with the ZBuffer. For anyone else having problems using a mixed set of render textures with and without MSAA, make sure the ZBuffer you are using matches your current render target in MSAA settings. I have now defined two ZBuffers, one for rendering with MSAA and one for everything else. Also make sure MSAA is disabled on your backbuffer, if you are rendering non-MSAA textures to it.
As InvalidPointer pointed out, I also had to copy my main scene render texture to another texture without MSAA using ResolveSubresource(). Then I can draw the new texture using a full-screen quad to my backbuffer.
I have one question though. Is there a way to render my MSAA-texture directly to my backbuffer? Or do I have to do convert it to a non-MSAA texture and then draw it as a full-screen quad as I do currently?
Edit: Okay I can answer my own question again
You can draw the MSAA texture directly to the backbuffer just fine. Just make sure you don't use linear filter or something if you are resizing the texture.