My tent has a rainfly, but I'm wondering if I need to carry a tarp for the extra rain protection. I don't want to pack a wet tent because that would damage the fabric, but I don't know if that is worth the extra weight of a tarp.
You do not need a tarp in addition to the rainfly of your tent (that's what the rainfly is for).
While it's always nicer to pack up a dry tent instead of a wet one, as long as you air dry the tent when you get home you'll have no problems with damaging the rainfly. If you do not dry your tent at home, it will mildew and smell really, really bad.
I generally set up a wet tent in a garage/basement and let it dry for 24-48 hours, or alternatively hang it up to dry.
I don't take a tent, I just take a tarp. Easier to pitch (if you have trees)
In 30 years of experience (4-6 weeks per year) I've only once gotten actually wet doing this. Old tarp. Heavy rain.
I pack the tarp last thing, to give it maximum drying time. If it's dry it goes in my pack. If wet, it gets rolled up in my foam pad. In the evening the tarp alone is set up as soon as my camp chores are done. It's generally dry by bed time.
I've also found that the ground under it is usually dry enough by then that my sleeping bag won't wick water. I'll spread my poncho first if the ground is really wet, then my foam pad, then the sleeping bag.
If I expect the weather to consistently wet, I'll bring a bivy sack too.
I don't take a tarp to protect my tent, I take it to create another dry area outside - typically for cooking and eating. It can also create shade for cooking, eating, and just lounging around. (On a rainy day I'll lounge around in the tent if anywhere, but on a nice day there are lots of options.)
Packing a wet tent won't damage it, but if your tent bag is snug you may find it difficult. Storing a wet tent for weeks or more might damage it. Your tent will get wet even if it doesn't rain, because of dew for example. So go ahead and bring a tarp, but not to put over your tent.