The motivation for keeping an application secret is to not create any problem with the current department.
There are no guarantees, but most places will respect your wishes for a while. But they may want to contact your references, for example.
They might also have to make a public statement if you are offered a job, depending on various rules.
A complicating factor is whether every committee member gets the word. And if they explore your candidacy with other colleagues (not always allowed) your wishes might not get communicated or respected by everyone. So, you assume some risk, but it is likely small.
And your situation isn't unique in any way. Lots of people need to do this, so most committee members are probably aware of it and have seen it before.
Let me re-order your questions so my answers can build upon each other:
Would search committee likely to respect such a request?
I have worked at multiple universities where we have handled job applications that we were requested to keep private. These job applications have been at multiple levels (from professor to upper administration). So, in my experience it is possible that search committees will respect these requests.
Is it reasonable to request keeping my application to a tenure-track position secret in the cover letter?
Given that it is possible, whether it is reasonable depends on your reasons for keeping your application private. The most common reason I have seen is because it would adversely impact a current position. But, there is extra work required for this, and sometimes it reduces the transparency of the application process. As many academics value transparency, there will be a need to balance the transparency of the interview process with your need for anonymity.
If at all possible, how long can such secret be kept?
There is no guarantee that a secret can be kept -- for most faculty positions you give a public talk which has to be announced. Someone might infer from this that you are applying for a position. If a faculty is large, someone might accidentally say that you are applying. (Since your name has to be announced to faculty so they can sign up to meet with you.) So, it is hard to put a precise bound on this.
Even if you don't write anything your application will stay mostly secret apart from reasons where it cannot be secret in order to be evaluated. Any place you apply to might contact your reference letter writers. If you get to the interview stage, this often includes an at least somewhat public lecture. If they want to evaluate your application these steps are mandatory.
Other than that your application is fairly secret by general procedure. The information you send in your application does not count as public so it will not be treated as such.
