Maybe people reading the comic books can explain this better, but in the movie The Incredible Hulk (the one with Edward Norton), Abomination is the enemy of Hulk, and at the end Hulk defeats Abomination, but after that, they don't say anything about him, even in the Avengers.
After the fight between Hulk and the Abomination, Hulk escapes, and they don't say anything about controlling Abomination. So what happens with him?
The movie universe has not explored this yet, although they probably will in future Hulk and Antman movies. In the comics universe, there are quite a few detainment options for the super villain on the go. They all have various plus or minuses but I'm going to highlight the ones that I think are likely fits for the needs of the movie universe.
The first would be Reed Richard's Negative Zone Prison Alpha. This is the most secure of the various prisons in the marvel comics worlds, but I find it the least likely to make the cross over since Fantastic 4 is still not a part of the Marvel movie fold. The prison is run by Richards and is located in a different dimension that should be impossible to escape from.
The Cube is slightly more possible, since it is run by SHIELD. However, the plotlines in the comics that have to do with that facility are ill suited to the movies. They may keep the name and lose the plotlines of moral ambiguous mind-controlling of villains.
It seems more likely that they would use something like the The Raft or The Vault which have no particular story baggage that go along with them. Both were superhuman prisons that lasted for quite a while, although both were eventually destroyed. They would have to hand wave some SHEILD or Stark tech in that would nullify powers, but they could do that with any of the movies.
The prison that I would say would be the easiest, and thus the most likely, would be Pym's The Big House. Pym basically created a shrink ray based off his Giantman research that allowed him to shrink down the inmates. Since shrinking is a cheap effect to do and it's instantly graspable by audiences without a ton of exposition, this seems like a perfect fit for the movies.
The final thing that could happen is based around the SHIELD TV show. Since we don't know the plot and SHIELD is typically tasked with taking care of metahuman threats, it is possible that we see the villains' containment options as a part of that show.
So, basically, something is being done with the bad guys to keep them from running around all the time and we have no idea what it is. Yet.
Agents of Shield mentions him briefly. Coulson uses guarding Blonsky's prison cell as punishment to Ward. Haven't seen The Consultant.
To answer the question "What happens to him (Abomination)?", according to "Marvel One-Shot: The Consultant", he is in some sort of prison. I don't remember if any other details are given.
Coulson mentions he is in the 'Ice Box' at some point in the AoS series.
I think he transformed back into his human form and was detained then and to control him from transforming again they probably put some sort of collar around his neck
