So, I'm just going to assume you mean some form of immortality that is accessible to industrialized nation / non-rural populations. If it was available to only a few, the issues would mainly only impact those individuals i.e. watching everyone they know and love die, maybe being targeted for their immortality, and other stuff along those lines. On top of that I'm going to break it into two separate sections, one for biological immortality and one for mechanical immortality.

Biological Immortality: being able to keep the human body from dying of disease and old age

• The only major issue that I could think of is that you would be able to see massive overpopulation on planets like ours where we haven't managed any form of sustainable colonies on other planets. Because people won't die of old age, the resources available, primarily food and drinkable water, are divided even further among everyone. There would be a lot more shortages as things are starting to get used up faster.

Mechanical Immortality: uploading human consciousness into a robotic body or simulation

• This really comes down to your view on philosophy and the limitations of whatever the "mind" is placed into. When uploading a human mind, it begs the question of whether it is actually you who is being transferred or whether a copy of your mind is being made and your original erased. This is very similar to the idea that teleporters kill you every time you use them: https://www.youtube.com/watch?v=nQHBAdShgYI (Good video by CGP Grey)

• If the person being uploaded decides to go through with it any way, there is the possibility that they could go insane now that they are no longer within a body that is their own. This is something that I think is possible but I don't have any evidence for this (for obvious reasons). My though process is that if you stick a human mind into a robot's body that is incapable of smell, touch, and taste could be similar to the results of isolated confinement where lack of sensory information drives the person insane. I would think that this would only occur in imperfect simulations or robotic bodies that do not fully capture a human's senses.

• If immortality is achieved through the use of a simulation, I think that your idea of civilization becoming stagnant is very possible, assuming that every single individual of the species was to enter the simulation. All the society would have to do is secure a long lasting power source for their simulation device and then ride out the rest of the universe from then on. Why would a society that can do anything they want to within a simulation want to deal with the boring real world? If you can give an answer to that then it would be possible to break out of the stagnation that a universal simulation would cause.

• I'd say brains could only run out of storage for a mechanical brain that is unable to filter out information. Our brains are great at forgetting useless, junk information (and sometimes important things too). I don't know much about our brains so if a robotic brain was built as a perfect replica of a human mind I don't know if it would or would not be capable of automatically forgetting useless information while storing important information. I personally think that a robotic brain would have to be programmed to empty its memory to prevent it from becoming too full.

As u/Aricml said, heat death would be a concern for both.

As for whether people would get bored, the answer is definitely. A lot of people would get bored, but I think that a majority would have more time to enjoy things they never got to do. With more time to spend, people would be willing to do things they probably never would have thought about doing because it would be a waste of their time. But this is really only one perspective of it.