This is something that my friends and I talk about often, and it's a subject of particular interest for me. In particular, here are some of the things that I find to be obstacles to the idea of living forever, or at least for a tremendously extended period of time:

-Population growth. Right now it's pretty well established that three generations will be in the world at any given time, with around 25-30 years between each generation. If we were to live to 150 (much less 750), suddenly that's five or six generations instead of three, potentially doubling the world's population, and that's assuming that people still follow the custom of having an average number of children between the ages of 25-35 and then stopping.

-On that note, if we are given the capability to remain sexually active for hundreds of years, then one person will be capable of multiplying into far more new people, rapidly increasing population growth. If we are not given that capability, then we'd all end up miserable as the period of our lives in which we have sexually fulfilling relationships becomes small.

-If we conquer diseases with such efficiency, then the tragedy of random, accidental deaths becomes greatly more pronounced. People may die from heart disease and cancer at a far reduced or nonexistent rate, but people will still die from car crashes, drug-related violence and suicide at the same rate they do now. Those deaths will become a lot more significant and difficult to handle, and we as a society will become either super-anxious and paranoid about the possibility of losing our lives or either perpetually in mourning (as our lives become longer we will have more connections and friendships and familial relationships, so we'll have more people whose deaths will affect us) or somehow immune or numb to the pain as a coping mechanism. The idea of war will become far more barbaric as suddenly instead of taking 60 years and a family away from a young man, you're taking hundreds of years and whatever else he could have done in that time.

-Any disease we are unable to cure with the potential to permanently affect people will instantly become far more terrifying. Non-fatal sexually transmitted diseases, loss of limbs or appendages, and debilitating mental diseases will be far, far more damaging given the greater lifespan that they will affect.

-Medical care will still, presumably, be as expensive and challenging as ever; however, we will have more people who need it and more reasons that those people need it. Most of these reasons will probably be expensive - for instance, if cancer treatment ends up being very very expensive, everybody will still want it - so our society is going to have to find some way to handle that. In a best-case scenario, most of the money we made would end up going to medical care, and we would never retire. The proportion would probably be far from that though, with 99% of people being unable to feasibly afford to maintain their own lives. Furthermore, we would need far, far more doctors, and therefore we would have to develop even more incentives for people to take that path in life and stay on it for as long as possible.

-Concepts that we have now (some of which I mentioned tangentially above) that are based on the idea of a 70-90 year lifespan and that we rely on will no longer exist. These include the eventual death of all dictators/tyrants, fairly consistent sizes in the nuclear and extended families, the age-based progression of opportunities and decisions, life sentences for crimes, marriage (would you want to keep the same spouse for 500 years?), retirement, etc.

That's what I think, anyway.