This is a good question and I have not seen it formally addressed in the amatuer race clubs I've run with.
It does not seem to be mentioned in the SCCA-GCR (other than a general statement about the tech inspector having a right to fail the car for any additional reason not mentioned)
SCCA-GCR
https://www.scca.com/documents/Club Rules/GCR2008.pdf
I'm sure if a tire is spotted during tech inspection with age-cracks, it would be rejected, but tires are often swapped after tech is done.
This may be less of an issue with autocross events.
There are "tire treatments" (such as "Formula V") that are supposed to rejuvinate old tires, but this suff is not allowed by the SCCA and most other road race groups. Might be used in autocross to dress up old tires though.
I'm sure NASCAR and F1 don't need to worry about a question like this, but I've seen some old race cars show up at some events with age-cracks and obvious dry rot. Sometimes, if it rains, some folks will put on their old "rolling around" street tires as rain tires....and some of these don't get teched since it's done on-the-fly.
I had a student show up at a race school at Pocono and two tires were so badly age-cracked that I wouldn't get in the car until he changed them....I guess the tech inspector missed them (I ended up giving him one of my spares and so did another racer).
Speaking personally, none of my race tires are older than four years and that's probably pushing it (and I "bag" them during the off season).