What are the requirements to receive a reward from a gym?

Solution 1:

To receive the defender awards, you must have at least one Pokémon in an allied gym.

To collect the rewards for having a Pokémon at a gym, you have to go to the shop while at least one Pokémon of yours is still defending at an allied gym. Thus it is best to capture as many gyms as possible (up to 10) and then collect before any get taken down. To get to the shop, from the home screen tap the Pokéball icon, then tap "Shop". Then inside the shop, there is a button in the top right where you can collect rewards for having Pokémon in a gym. Note that you can only collect once per 21 hours.

Also, as pointed out by Virusbomb, "the rewards for defending a gym are 10 Pokecoins (which can also be bought with real money and are used to buy in game items) and 500 stardust. You can get a max of 100 coins and 5000 stardust every 21 hours with Pokemon in 10 different gyms."

Solution 2:

UPDATE: Apparently, Niantic has taken notice of everyone's complaints about the slower rate that coins come in from defending Pokémon. There has now been another change to the rate at which defending Pokémon earn coins. Instead of 1 coin per hour, it is now 1 coin per 10 minutes up to a maximum of 50 per day. From the Silph Road subreddit:

Update: There have now been 2 silent changes to coin accrual in the new system:

Coins are now granted 6x more frequently: in 10 minute increments, rather than hourly

Gym coin earnings are now capped at 50 coins/day

The new limit was confirmed on Niantic's support page, but has since been removed. This is what it originally said:

You redeem your PokéCoins only after your Pokémon returns from the Gym. You can redeem a maximum bonus of 50 PokéCoins per day. If you have more than one Pokémon return from Gyms in a single day, the maximum bonus for the day is still limited to 50 PokéCoins.

Regardless, the maximum is still 50 coins per day, which means that after a Pokémon has been in a gym for 8 hours and 20 minutes, it will no longer earn any coins for you.

As of the recent update that completely overhauled the gym system, the length of time a Pokémon is in a gym determines how many Pokécoins you are rewarded and you are given this reward all at once as soon as the defending Pokémon is returned to you.

Based on the observations of players here, it appears that there is a minimum amount of time a Pokémon needs to be in a gym before it will give any rewards upon being removed from the gym. From the comments there, it looks like the minimum amount of time a Pokémon must be in a gym is 1 hour, at which point, you will be rewarded a varying amount of Pokécoins starting at just 1 Pokécoin when the Pokémon is returned to you.

Having one in for over an hour still results in no coins.

*edit: GOT MY 1 COIN, after crashing/restarting the app. must have been a glitch.

Mine was in the gym for over an hour and I also fed him a berry before being returned. ... just got my 1 coin

My snorlax was kicked out and came back with one coin only

How long approx was it in for?

A little over an hour I think.