The movie is about a couple where the wife/girlfriend gets sicker and sicker. I can't remember her diagnosis, but I think she gets Alzheimer's or some kind of amnesia and sinks into a more and more "distant" state of mind. Her boyfriend/husband tries to keep her focused, but it is difficult.
I think the movie was made in the late 1990's or early 2000's, maybe later.
It is not "Amour".
Possibly Se souvenir des belles choses (AKA Beautiful Memories) from 2001:
Although barely 30, Claire believes she is showing the first symptoms of Alzheimer's disease, a condition from which her mother has recently died. Her sister, Nathalie, is certain that her memory loss, caused by a lightning strike, is temporary. In the clinic where she is being treated, Claire is attracted to Philippe, a man who is still traumatized after a car accident in which his wife and child were both killed. In spite of their personal tragedies, Claire and Philippe fall in love. When Philippe recovers, Claire moves into his home. Then Claire's condition takes a turn for the worse...
Here's the trailer:
