I applyed tung oil to my daugher in laws car tires for a shine in Tucson, AZ My son claims that the oil caused the tires to crack. He says I owe him a new set of Michelins. I disagree. What do you think?
I think a new set of tires seems pretty t=rad. Of course you could always pro rate the tires on mileage. I have some michelans that look like crap. We have an old driveway made for skinnier cars, the blackwall is worn to have ugly white spots where it has worn off. Only a cosmetic issue they say, and no trouble to the tire integrity. Gotta get past the next tuition payment for daughters college, still got enough tread but hope to make it through winter, kicking in 4wd as necessary. Now the question is what do you owe. Tires are tough and I don’t think what you did caused any damage.
Give him a choice.
Inheritance or new set of tires?
My Michelins seem to crack a little too. It’s because they last soo much longer and see more uv and elements. last set I had 120k miles and still had 3 mm when I removed for wet weather safety. Light cracking is normal. Buy him tire covers for when it’s parked to slow down the uv damage! 
Whatever you decide, you might want to stop with the tung oil.
Why were you putting tung oil on tires in the first place? Were you asked or instructed to? Anyway, how old were the tires on the car?
Tires have waxes and anitoxidants (AO’s)in them to slow down the natural oxidation breakdown of rubber. The waxes form a barrier and the AO absorb what gets through. Anything that dissolves the waxes is going to allow the oxugen to penetrate faster.
Petroleum products should NOT be used on tires. Many tire dressings have such products in them. Remember, if you get a black mark on your hand - it’s bad!!! (That is not to say that no mark would be OK.)
Tung oil is frequently diluted with turpentine or some other solvent. So the use of tung oil on tires is one of those “DO NOT”'s.
Frankly it was not a great idea. It is not a good idea for your car either. I recommend not putting anything on tyres, but if you feel you must have shinny black tyres, then use a product made for that use. Don't expect it to extend the tyre life, but it likely will not do much damage either.
So the auto part of the question is resolved. Now it is time for the people part. You son needs to learn how to be a man. That includes understanding that people make mistakes, including fathers and sons. Your mistake was using that stuff, but I would guess you use it on your own tyres so you certainly was not trying to do any harm. You son needs to learn S*** happens. It is very minor. I suspect your daughter in law appreciated the effort. If so be glad that your son found a good wife and hope he will grow up.
Have him read the replies to your question.