There's only one thing you shouldn't do, and that's store them wet in a metal box. Normally, I'll let flies dry on a foam patch in the boat, sling pack, etc. Hang 'em on the visor of your truck - that adds tons of style and mojo to your fishing karma.
Let them air dry so that the hooks and/or box does not rust. Unsure putting them on your visor does anything except keep you from being fly free when you forget to load or pack your box...then it is a trip saver!
My overhead in my truck is loaded with fly's that I have used and cut off. If I'm sitting in my truck and wonder what to use I just look up and choose. I have dries, nymph's and WB's stuck there. All barb-less of course.
About the only way you end up with rusty hooks is to leave them soaking in Water or fish in the Salt. I usually just put them back into my box if I'm fishing in fresh water. They never seem to rust on me.
For the day: I put them in a small empty pocket of my vest. One with a mesh front is even better. That at least starts them drying. At home I put them into the coils of my tying lamp's support spring, or in the dust jackets of shelved books, or just loose on a table, overnight. Before putting them back in a box, primp them with your fingers, like your mother straightening out your cowlick. Untangle the feathers and twist the wings back to proper orientation. Resharpen points if needed.