I'm looking for a Bradbury story in which a character has to report for euthanasia because he has hit the age for it. He has a conversation with his son, and we only gradually understand what's about to happen.

I can think of 3 or 4 stories involving a similar plot, but not any by Bradbury.

I believe the short story you are looking for is "The Test" by Richard Matheson. It takes place in the future, where all citizens over a certain age are required, once every decade, to pass a test. If they do not, they are euthanized. I first read the story as a teenager, and was tremendously moved by it.

tgd:

Could you be thinking of Logan's Run by George Clayton Johnson and William F. Nolan?

It also sounds quite like the chilling 'Examination Day' by Henry Slesar (? - not sure if that's the writer's exact name), except in that story it's the boy who undergoes the test on his 12th or 13th birthday. It's not until the end that the true nature (outcome) of the test is revealed.
A stunning little story, but alas, not by Bradbury.