Do cars age?
Yes. There are parts that need replacement regardless of whether they're actually used. Mainly rubber and some plastic parts. Rubber will crack or rot, and plastic can become brittle and crack. The most common stuff to need replacement from age alone is weatherstripping. Some electrical insulation could also start to go, but that will likely take a lot longer. Tires weaken and crack, too.
Also, anything that can rust, if you live in a humid and/or salty environment.
If you live in a desert, all you have to worry about is rubber and paint, though.
Oh, and as long as you drive it every couple of months (or better yet, weeks), you don't have a lot to worry about with regard to everything else. Leaving them sit does pose other problems, but I presume you're just talking about driving the car on the weekends and such.