In my own experience the vast majority of "career First Officers" have been passed over for promotion by the airlines management training department, rather than have expressed a desire not to be promoted. Usually the reason is that the candidate is assessed as either not having the requisite experience at that point in time, or assessed as unlikley to stand a reasonable chance of success if put forward for a command course. When this occurs, it will usually result in significant dialogue between the two parties to establish an understanding for the decision taken, and a roadmap for future improvement to meet the criteria for consideration.
There are many reasons why people in this situation do not advance further to a successful command upgrade. In some circumstances people who are perfectly able and capable in their everyday performance, simply fail to cope with the stress they feel when under an artificial testing regime. Similarly people often fail to show sufficient assertiveness when required. None of this makes the candidate any less of an individual or in most cases any less of a perfectly competent and able pilot. However if the problems cannot be sufficiently rectified, it will prove a bar to further selective advancement to the role of Captain. In some cases there is an acceptance of this and the individual is satisfied to accept their role of "career F/O." In my experience these people have the obvious advantage of being the most mature and experienced colleagues you will fly with. The wealth of experience they bring to the operation is very much a safety advantage, and they undertake all the roles of second in command with confidence and aplomb which makes for a relaxed flight or trip. Although they haven't been promoted, they understand the role of the other seat very well, and are able to be much more pro-active and understanding of what is required to make the captains job easier.
Given the different structure and consequential lifestyles that different airline operations afford, it is also going to be inevitable that some people will elect to hold themselves back from promotion for a whole gamut of lifestyle reasons. Where this does occur, it is likely to be in operations that involve structured short and long-haul flying, as well as those with a geographical diversity of bases. Despite this, most people in a competitive career will for all the obvious reasons, always aim for the earliest opportunity for progression. Not least because this may involve issues of long term seniority, pension, salary, and uncertain windows of opportunity. However as in any walk of life there will also be people who have personal issues in their own lives that they feel are more relevant and preclude the desire to advance to captain.
I have to say that I have never come across a situation where an airline would seek to terminate somebody because they either didn't want to be promoted, or because they were assessed as unsuitable for promotion. Certainly employers will normally employ those individuals who at the time of interview seem likely to advance to the role eventually. However as long as the person maintains an acceptable standard for the job they are preforming, I would have some serious doubts that any such practice was lawful in most countries, and certainly it would be grossly short sighted and undesirable.