You can buy fancy stationary / résumé paper that has a percentage of cotton fibers in it. I've used that on a few bills I've printed out myself. Frankly the difference isn't huge, at least not to me.
Otherwise, there are other papers that have an off-white color, or subtle colored fibers in them. It depends on the note you're trying to replicate.
I have a few prop bills (Now You See Me, Elysium) that are just printed on apparently plain white paper. Texture, weight and feel would be irrelevant on screen, so I presume they're simply printed as cheaply as possible, depending on the quantities required.
-MJ