Yes, they go hard with age. How long it takes, depends on how they're stored: outside in the sun is the worst. Inside a nice cool, dark shed is the best.
I mainly notice old tyres on hard-pack, where they don't bend and conform to the ground like they should, but even in good conditions there is a (barely) noticeable difference compared to a fresh version of the same tyre. They generally feel a bit more wooden, and give up a bit earlier and more suddenly.
When they're really past their best and are a lighter grey colour, they tend to feel a bit chalky to touch, and they wear out really quickly (like chalk, I guess). But if you can't pick a tyre like this as being well past it's prime just be looking at it...