I could write 1000 page book on the logical social problems that would cause without even trying very hard.
Besides the massive overcrowding (which we could actually engineer around) you are going to end up with a society you wouldn't want to live in.
If you think the rich are rich now and the poor are poor now, then give them essentially unlimited lifetimes and see what happens. The 1000 year number is common in discussions about being ageless because that is about how long you should be able to go statistically without dying in an accident.
No one born after the first 50 years this is introduced will EVER have a job better than minimum wage, and they will be lucky to get that. It doesn't matter how good you are, someone else is better, and has 50 more years of experience, and has had his position 50 years. Not that there would be any jobs to apply for in the first place.
Oh, and I hope you like paying taxes for welfare. Take our current poor population. Make getting any jobs impossible, now make them immortal and fertile the whole time.
Essentially our culture isn't even close to advanced enough for that kind of technology. History has shown time and time again what happens when advanced technology is introduced into cultures that are too primitive for them.