Elysium is a fantastic concentration of wealth in the ultimate walled community. I don't recall specific population numbers, but it is clear they represent far less than 1% of the human population. The people of Elysium control the means of production and police forces. Unemployment is rampant and they use Earth as a pool of cheap and very disposable labor.
The ending of the movie, where everyone is made a citizen of Elysium and receives health care, is a stopgap measure to solve the focus of the movie: healthcare. But, as the question correctly surmises, this isn't going to last long. Elysium's health care system designed for the small population of Elysium cannot possibly keep up with the entire population of the Earth.
In the movie's simplistic solution, somewhat realistic given the desperation of the characters who kicked it off, Elysium will likely rapidly exhaust its resources attempting to cure the world's problems alone. This might leave people slightly better as it eliminates Elysium as a brutal, exploitative superpower, but it doesn't do anything to alleviate their basic economic and political problems.
Even if Elysium could sustain this, lowering the global mortality rate and increasing the average human lifespan without other changes would exacerbate the existing population and poverty problems. Our current population explosion is, in part, a result of the death rate dropping faster than the birth rate along with a greatly increased life expectancy.
At minimum, the birth rate would have to be lowered to meet the new lowered death rate and raised life expectancy to avoid a population boom.
Looking a bit further down the road, Elysium's magic cure-all machines are only a band-aid on the much larger health problems of extreme poverty. It would be better in the long run to treat the cause (extreme poverty) then the symptoms.
To truly effect real long term change, the means of production and wealth accumulated in Elysium would have to be distributed differently. The world in Elysium is highly automated with sophisticated robotics and technological advances. This high level of production efficiency means it is possible the entire population of the Earth could be sustained in comfort, but it is unlikely that enough jobs could be available to give all humans meaningful work and distribute the wealth.
Instead, wealth would have to be distributed through some other means. Guaranteed Minimum Income is one such possibility.
Elysium is, of course, an allegory. These are all exaggerated problems we're facing today. Like today, rather than dealing with the big, long term problems, the filmmakers chose a short term fix to wrap everything up for that neat Hollywood happy ending that leaves you wondering.