When I saw the movie, I thought it was fairly explicitly saying that the future could be changed, and the future was not fixed.
Thinking back, I got this mostly from the line of dialogue where Louise asks Ian if he could know the future (or see what was to come, some wording like that), if he would change anything - and he said something about maybe making sure to tell people things, let the people he cares about know, for what looked to me like an implied admission that he wouldn't try to change events, just maybe his reactions - though given the content of some of the flashbacks, perhaps he didn't mean it quite as much as he thought he did.
Along the same lines, I thought that was the actual reason for the fight with her husband - that she could change things, but had chosen not to. Perhaps she could not specifically choose whether her daughter was born with that disease (not human-changeable), but she likely could choose whether to have a child, or with whom (and thus which genes might get passed along)... and she chose to have a child knowing what would happen, because she chose to spend what time she could with her daughter - since she knew about her from the visions, and would mourn her either way, and chose to take the bad along with the good of her life.
And similarly, the setup with the General Shang looked like it worked the same way - in the flash-forward, she has to get the information from General Shang pretty specifically, in order to reach him and change his mind in time in the past. The dialogue suggests that if he hadn't given it to her - that is, if he hadn't, in the intervening time, changed his mind and agreed to change his own mind in the past, the information wouldn't have been there in the past for her to change his mind with. I thought it significant that it had to be him explaining and agreeing and giving her and in, rather than some outside account of how she called and what she said where we don't know where the information came from (which might have suggested a more deterministic viewpoint, if the info was simply "there").
I saw the future she was seeing as a choice she had made - one she could walk away from, but chooses not to, one she could work towards, and decides it's worth the pain. What she loses is the ability to not look at what might-have-been, she knows what she'll be walking away from pretty specifically when she makes that choice. Just like the general made the choice to give her the information to change his mind in the past. Just like her husband blamed her for not making a different choice when she knew hers would lead to pain.
So, to go back to your question, no I don't think she's immortal or in a closed time loop. The future is being fixed through her actions and choices, just like everyone else's, as time goes along. She can revisit memories, but that isn't the same as living (memories usually aren't), especially since she can only change what she (will) do the first time around. And she will continue to live, to make choices, and eventually die.