How would I go about finding out the licensing rights for NeHe's TGA texture loader? It works very well and I'm considering using it in my games.

IANAL, but the original tutorials were written to be used freely, so you should be fine. The NeHe readme says "If you use this code in project of your own, send a shout out to the author!" so I think as long as you name the authors of the tutorial (normally Jeff Molofee combined with another contributor), you should be all good. That said, the original tutorials didn't contain any real "license" for use, so I guess they are a legal grey area, although it's highly unlikely one of the authors is going to cause a fuss over it.

When we release the new set of tutorials we will make sure that the license is specified and liberal. For your specific use-case you might be better off using the SOIL library which is public domain, easy to use and supports loads of formats.