This article says that Running shoes only last about a year regardless of mileage. Is there a similar time frame for climbing shoes? I searched the internet and it seems that most climbers wear through their shoes' soles well before they might age out, but it also seems that every climber who responds to such questions climbs 2-3 times a week or more.
Climbing shoes don't wear out from age alone (in any realistic time frame that is).
However, the rubber in the soles do age (probably because it "dries" and oxidizes over time) which affects the shoes performance negatively.
You can fresh up old but not worn out soles with sand paper or a steel brush to get some of the original stickiness back. And of course you can always resole.
Good fit and sticky rubber is what makes a good allround climbing shoe.
They won't wear out from age alone. It seems like the big sticking point to the author of the article was that the foam loses its padding over time,
As a general rule, he tells customers who are casual runners they can go about a year (even if you don’t rack up 300 to 500 miles in that time, the foam still loses its oomph).
That really doesn't apply to climbing shoes, especially since you can get your climbing shoes resoled once the rubber wears out.
It also depends on how hard and how often you climb, for a professional rock climber with sponsorships doing hard routes in Yosemite, it would probably be worth it to replace shoes fairly often, but probably not for most people.
Personally, I have done almost all of my climbing in shoes that were 20+ years old, and while new shoes would probably be better, I haven't bothered to spend the money when the old ones work just fine.
Rubber oxidizes. The term (in time) that your rubber oxidizes at cannot be said because of certain conditions, but I can name a few tips to help prevent that from happening.
Avoid heat, UV exposure, oxygen, and pressure. Of course, these cannot be completely avoided as you rock climb or do anything with your shoes, but being conscious of how you store your shoes makes a big difference over a long term. I have also heard of people seran wrapping their shoes over the winter.
From personal experience, I've noticed my shoes' rubber get stiffer after 3 months of no use. But it also depends on the shoe and if the rubber is meant to be stiffer or softer.