A common concern about life extension is that we would rapidly overrun the planet, and without management, this could happen. However, there a few factors in our favour.
In many countries, fertility rates are already dropping below that required to maintain a population, and without immigration the number of people living there would be falling.
Even with biological immortality, if people were limited to one child each, then there would be a finite maximum population to deal with. And biological immortality does not mean fictional immortality - people will still die in accidents, conflict or by choice, meaning the global population would eventually start falling again.
We will have a lot of time to come up with a solution - the population won't double overnight, it might take a 100 years or so, and it's impossible to imagine what technologies may have been developed by then to help.
See also: Space Colonization
Overpopulation Blog Posts
Crude birth rate of England and Wales hits a new low
BBC - 01-Aug-2019
Total fertility rate is lower than many other years except 1977 and 1999 to 2002Read more...
China child restrictions leading to fall in population
Futurism - 18-Jan-2019
More countries are having to plan for increasingly aged populationsRead more...
Photosynthesis fix boosts crop growth by 40%
Independent - 04-Jan-2019
More efficient photorespiration pathway could result in greater crop productionRead more...