I've carried, and used, a spare tire. There was a wire in the sidewall of my touring tire that broke and caused 5 flats before I found it. I put on the spare tire and, enraged, walked the old one down the road 50 yards to toss the old one in somebody's garbage can.
The replacement wore out, but I was able to baby it 100 miles after it flatted and I noticed the cords showing through the tread until I could buy a new tire (great LBS in West Yellowstone, FWIW).
So it's certainly possible to need a new tire on tour. How do you decide?
Standard tire sizes (700C and 26") are fairly easy to find if there's a bike shop nearby, and (for the 26" size) at Wally-world. Other tires (650B, for instance) are going to be hard to find. Can your schedule and finances order a tire online, FedEx delivery, and you sit and wait for it to arrive if your tire can't be fixed?
How far are you going? How heavy is your loaded bike? How hilly is the route? I wear most tires out in 2,000 miles. Some tires (Continental Contact, Specialized Armadillo) last 2-3 times longer. My daughter, the featherweight, rode used tires all the way across the country with me and for another couple years before hers needed to be replaced.
How isolated is your route? If you're riding well-trafficed roads between good sized towns or cities, you can probably get a lift if you need one. If you're riding little-used dirt roads where traffic is measured in vehicles per month, you might want to carry spare tires.
How diligent are you about checking tires? I tend to ignore tires (except for pumping them) until I have to look for what caused a flat. If you are starting with fresh rubber, not going more than perhaps a month, and checking the tires regularly (so you can get a replacement shipped ahead if necessary), there's less cause to carry a spare.
Finally, have you considered carrying lightweight folding spares? Some models are three times the weight of others. Consider taking a light tire, and if you have to mount it, order a new heavier tire or buy one in the next town and swap them back out when you get the sturdier model.