Unfortunately, you can't rent TV Shows from iTunes anymore[1].

[1] http://allthingsd.com/20110826/apple-pulls-the-plug-on-tv-re...

I'm not sure why iTunes rentals didn't take off. It seems like most people do want rentals, since they watch a show only once. Maybe the price point wasn't very compelling? Or the selection of rentals wasn't as good as that for buying?

I don't think it was the selection of rentals, but other reasons including: Apple TV wasn't as popular as it is now [1]; and, most likely, you're already paying for TV Service with your internet service provider - so why you would pay (again) for a show you've already paid for? (And, most likely, you're paying for DVR service, so you could have just recorded your show for future (and, potentially, unlimited) viewing.)

1. http://daringfireball.net/linked/2012/07/24/apple-tv-xbox

I wonder if that means Redbox and other physical renters still have a chance of staying on.

I feel that if renting media still exists in the future, it would be a solid niche for physical media to exist for. I always imagine that one is more likely to watch Netflix DVD's that arrive in the mail, because it exists in physical form and are more of an "incentive" to customers to continue with their experience. Whereas I always see my friends with Netflix's streaming library on endlessly scrolling through it, unwilling to commit to watching any particular film or show. Is there a random function?

A physical rental is like an investment that you've already committed to.