How to smooth PLA Prints without using sandpaper, solvents or paint
Unfortunately, there is no way to smooth PLA without sandpaper, solvent or paint - but you can fix the finish after sanding.

If you heat the plastic after sanding to just the point the outer shell starts melting the horrible matte finish goes away and the original color of the plastic returns.

You have to heat the plastic and then remove the heat source just as it's start melting (right before the original color returns, so when you see it working it's too late) because otherwise the object will deform.

I use a heat gun set to 180C and work quickly in short bursts, an hair dryer is probably nowhere as hot, so it will take longer to heat the surface.

You have to do this after sanding, applying enough heat to smooth the layer lines will just cause the object to melt and deform.

Try on a few failed prints first, you will need to get the feel for when to stop heating and you will still probably ruin prints every once in a while. 

Another option is to coat the object in an epoxy that will hide the layer lines the best known brand for this is XTC-3D.

If you do decide to paint, you get extra thick primer that's supposed to fill the gaps between the layer lines so you don't need to sand as much (sorry, can't remember the brand name)