I have been in a tenure-track position in 5-6 years and I am applying to positions in other universities, including tenured positions. For the list of references, should I include my former PhD advisor? I have some other good names to include and also I have a good relationship with the former advisor. I was wondering the pros and cons of including/excluding the former advisor in the list of reference.
This might vary from field to field, and based upon how established and well-regarded you are within your own field, but personally I would not feel that it is mandatory to get a letter from your prior advisor, if you have other letter-writers who you think will be more suitable (know your work better, are better-regarded in the community, will write you a stronger recommendation). At this stage in your career most faculty are now established enough that they are your own brand and can stand on their own, separate from their advisor. In particular, 5-6 years in a tenure-track position is probably far enough along in your career that I don't think the hiring committee will look askance if you don't have a letter from your prior advisor.
In any case, if the hiring committee wants an assessment from your former PhD advisor, they will ask your former PhD advisor. For jobs at this level, it's not unusual for them to ask others for their opinion of you (beyond the letters that you provide), if you are a serious candidate.
That said, usually your former PhD advisor is someone who knows you well, wants you to succeed, appreciates your work and your interests, and is well-informed about your research -- so they are often a good choice of a letter-writer, all else being equal.
My impression is that at this stage it has much higher value if your recommendation letters can prove that you already somewhat established yourself in your field, your independent research topic is going well and people beyond your lab/institute start recognising your name.
Most cases a recommendation letter from your former supervisor hardly can demonstrate independency, it may even suggest just the opposite. I saw cases when a well-meaning ex-supervisor was writing pages about how friendly and docile is the candidate, how good student and doing always everything as ordered. This kind of recommendation letter you really DON'T want to get for a tenured position.