How long is it safe to store new tires before they degrade in quality? I bought some road tires that were made for flat resistance, but within two months, the sidewalls on both got damaged to the point where I was getting repeated blowouts. I wrote to the manufacturer, and they were kind enough to send me a replacement pair. 

In the meantime, I bought some new flat resistant tires to replace the busted ones. 

Now I have two new pairs of flat resistant tires, but I only have a use for one at a time. How long can I store a new pair before I have to worry about the material degrading, dry rot, etc.?

(not mentioning the brands here because I don't want to bash the manufacturer that sent me the replacement.)
It depends on the storage conditions. If kept at room temperature, not excessively humid or dry, and kept in the dark, they can be stored for a long time. Oftentimes for years. 

Your best storage bet is to wrap them in plastic and stash them in the back of a closet. I've kept high-end tubulars for two years that way and that rubber is much more sensitive to storage/dry rot compared to puncture-resistant tires.