Mainly behavioral problems -- the worst extremes being things like cannibalism and guarding the feeder and waterer so that the low-rank birds starve.
If you look at the hatchery notes, breed that say "tolerates confinement" do better in a tight situation than breeds that say "good foragers" or "good for ranging".
Lack of space isn't a guarantee of issues, but when issues arise it very often seems to be a contributing factor.
Yes there may be behavior problems and stress. "Tolerating confinement" or keeping chickens in one of those cute little coops without outside time IMO is not much better than those chicken production buildings where they claim "cage free" but the chickens are crowded into this big area without cages. Those chickens also have issues and most require cutting their upper beak really short so they can't pick on eachother.
I would say that if your chickens free range frequently, they can have a smaller coop and pen. Mine free range most days in my backyard and my pen is 16x16 for 15 chickens. The coop is fairly large. My pen is built like Fort Knox, so they live most of the year with the coop door open. In the really cold weather they are in the coop Or in the coop with a heater (because I can) that keeps the 3 water buckets unfrozen, about 30 degrees.
When I lived in Florida, my chickens did not have a coop, just a covered pen and a rubbermaid tub on it's side for laying eggs. It was in the shade all day. I did nothing in the winter.
What I'm saying is you have to think of the chickens' situation, if there out more frequently then their coop and pen can be smaller. Use your judgement and if you read what others do, use that as reference NOT gospel. And remember they are animals. If they get dirty feet well they will. I have silkies with dirty feet! If I hold them I also own a washing machine. I snip their butt feathers, worm them, give them veggies and treats, and physically handle each one weekly to check their weight. And I love them all !
One of the most interesting tips I have is if you get mulch, buy a color that's close to the color of chicken poop.
Sandy rocky silty crap. Im constantly pulling rocks out as over the years it slowly gets better to grow in. By occasionally throwing down a handfull of seed where they like to root and dust bathe, they'll dig the crap out of that to get the seeds and dig up a lot of rocks, which are now on the surface so a quick drag of the rake and I got them all out.
, he said this is the first coop he's been around that doesn't smell bad. All I do is pick up the poop every morning with tongs, spread out more pine shavings and newspapers, use fly strips, use fly pheromone product, put out clean food and water in dishes. Rake run regularly, cleanse and sweep coop regularly. Use DE in their dust bath area, which they have to have. They will always eat all grass in their run/coop area; its programmed into them to forage and peck and scratch. It gives them pleasure. But picking up their poop daily will prevent smells. Also you can grow herbs and put them in their bedding; very nice for them and you. You can put some hay down where the ground is bear and that looks more attractive and the girls love to rummage thru it. 
