I have to humbly disagree with Sheldon Brown. Old dry rotted tires that have hard cracking sidewalls need to be replaced. I have had a lot of tires just explode when the tire sidewall failed and the tube burst through.

The weird thing is, age does not seem to matter as much as the environment the tire was stored in and how much it was used. A tire that never gets rode will become harder faster than one that gets some use. I have seen badly dry rotted and ruined tires on nice bikes from the mid 1990s that didn't get ridden much and I have seen tires from the early 1980s that I would still trust. Maybe is has somewhat to do with the initial quality of the tires or maybe older tires were made better.