Once you have tenure, presumably you're never going to have to search for a job again. However almost all professors have a CV on their website, and fairly recent ones. Why? They already made it.
Upon joining the club of those who've "made it," you find that there are further gradations. A tenured professor can safely breathe a sigh of relief for at least a year or two, but eventually she will notice (i) that her salary is lower than most of her tenured colleagues and (ii) that she is still being asked to document and defend her productivity across several axes: research, teaching and service. (Also, in many cases, (iii) She is increasingly pressured to apply for and get major grants. In many cases, her future salary and promotions depend on this to a large degree.)
The main tool to exhibit your recent accomplishments in research, teaching and service is...your CV, usually. In fact, "whether they want a good raise or not," most faculty are required to submit annual "activity reports" in which a CV is often a required component. (When it isn't, it usually means that we manually copy information from our CV into some form or type it into some internal webpage. So having an updated CV is more or less necessary for many faculty members.) Certainly having a carefully groomed CV is necessary when you go up for "promotion to full", apply for an endowed chair or distinguished research position, and so forth.
The bottom line: academia has bureaucratic, competitive and hierarchical aspects all the way through. Most people who have made it tenure in recent years have learned how to keep reasonably up-to-date CVs without too much trouble or heroic effort -- the CVs do not need to be rewritten from scratch to showcase their skills for some new job, usually, so it is mostly a matter of regular updating -- so honestly, this is not the worst part of the bureaucratic minutiae.
Some reasons that I've needed to have an up-to-date CV:
Applying for promotion to full professor.
Applying for a funding for a sabbatical leave visit to a research center.
Applying for grants. (Most funding agencies have very specialized formatting requirements for shortened CV's. For example, you're only allowed 2 pages for the one you submit with an NSF proposal.)
An external review of our department and its programs.
It's such a common need that I just make a point of keeping my CV up to date as new publications and other events happen.
Everybody needs a CV to help others better know them.
I am not a professor, but I think even tenured professors may maintain their CVs so as;
- As an advising professor, they need to hire or accept research (PhD or masters) students,
- As a university professor, they may hire PhD graduates in form of postdoc positions,
- As a person working in university or industry, they need collaboration;
and for many other reasons.
They need an updated and maintained CV so as people who want to collaborate with them in a research project get know them, their research interests and their previous works or at least their educational background.
Research students need to read their CV to get familiar with their research interests and know their past PhD or masters students' projects. This helps them to better choose an adviser whose education and research background is so near to their academic interests.
These professors may also want to apply to get a project, they need a complete CV to introduce themselves to the project owners or investors.
I'm not sure if this is already covered by the existing answers, but one use for an up-to-date CV is: you may be asked to write reference letters for junior researchers seeking tenure or promotion or a higher-ranking job elsewhere, or act as a referee on someone's grant application. It is then part of the bureaucracy, and also informative, for those who wish to solicit letters from you to be able to see what you have been doing.
Note that this is not just a matter of looking up publications: a CV might include various service roles such as administration within a scholarly society, plenary talks and invited courses, and so on.
Great question! In addition to the formal, bureaucratic reasons (evaluations for grants, promotions, jobs etc.) mentioned by the other answers for why it's good to maintain a CV, and preferably a public one, let me add another, informal, reason.
In academia as in many other areas of professional endeavor, there is a kind of "the rich get richer" effect (a.k.a. a virtuous cycle, or positive feedback loop): once you are successful enough, you can leverage your successes to get more success, and the CV (and other similar forms of seemingly gratuitous self-promotion, like maintaining a web page) can play a role in reinforcing this virtuous cycle.
Here's a simple example (one among many - one can easily think of several similar situations that illustrate the same principle): I am occasionally invited to speak at conferences. The conference organizers don't have to invite me, and when they are considering whom to invite, doubtlessly they have various other ideas about possible speakers. Now, imagine that they consider either inviting me or another researcher who does work which is about as good as mine, but does not maintain a web page with easily accessible information about their research, or a CV. It is easy to imagine that all other things being equal, I will appear as a more attractive potential speaker and end up being invited, and the other researcher will not be. Then, when I eventually travel to the conference, all sorts of additional good things can happen - I may impress people by giving a good talk, and make it more likely that I can get more speaking invitations in the future; I may form new collaborations, or see an interesting talk that gives me a good research idea, etc. -- all of which are things that are likely to breed even more successes for me in the future. The virtuous cycle is reinforced, and all (hypothetically speaking) thanks to a humble PDF document of several pages that I posted online, that I can easily maintain with a minimal investment of a few minutes per quarter. Of course, we shouldn't forget that the work one does the rest of the time -- all the stuff that ends up being mentioned in one's CV -- is a lot more important and essential, but in a simple cost-benefit analysis, it's easy to see that maintaining a CV is a no-brainer.
Finally, let me also mention an unrelated psychological observation, which also answers your question but in a kind of indirect way: because the competition to get tenured professorships at good universities is so tough, this causes a selective pressure that ensures that many tenured professors are highly ambitious, competitive people. That means that they are not the sort of people to think "I've made it, now I can relax and play golf for the rest of my career"; instead, after getting tenure, many of them look ahead to see what more they can accomplish. I can assure you there is a lot more one can achieve in life than being a tenured professor. So, as others have pointed out, your basic premise that a tenured professor represents someone who has "made it" simply does not fit the psychological profile and mindset of a lot of tenured professors.
There are two wildly unrelated questions here: why keep an updated CV, and why put it on one's webpage (if one has such a thing). As in other answers, there are many reasons to have an updated CV, if only the immediate one that most (e.g., math) depts require an annual report, which is a sort of edited-down version of one's CV to include only recent things.
The question of why put it on-line is very different. Much of the information in the type of CV demanded by department heads or deans is surely of little interest to people interested in the subject, as opposed to the sociology of the milieu, or gossip. One can imagine that people are proud of prizes or invitations and such, and, indeed, this does impart a certain sort of information about the esteem in which their work is held.
But, equally surely, the essential information on a web page is the papers themselves, not the life story or prizes of the person. That assumes that the goal is making information available, as opposed to self-promotion.
So, for example, of course I have an updated CV that I prune to make my annual report, but I don't put this on-line.
This question reminds me of the parolee who resumes his life of crime as soon as he steps out the prison door. Tenure should be seen not as the end of something but the beginning of something. Everything before tenure was an effort to demonstrate how one would be once tenure is achieved, and to quit just because one has finally been admitted irreversibly to the team, betrays the team: your colleagues, the school that hired you in good faith, and more vaguely but more importantly, the field that you've committed yourself to studying. OK, I realize you're tired, you've made a superhuman effort, but that's why you've been granted such an extraordinary position. Take a sabbatical, get over it, and get to work.