Should I simultaneously apply for multiple jobs in different ranks at the same university?

My experience is in mathematics in the US. My personal experience is that people will not read anything especially deep into applying for both jobs. Everybody understands that the market is tough, and basically expects everybody to apply to every opening. I've seen instances where a person was considered both for a postdoc and TT position at the same university, and this was not seen as an issue at all (there was much more strategizing around the question of whether they would come for a postdoc).

In the vast majority of cases (without, say, a severe geographical constraint), I think if you consider TT and postdoc at the same place as both reasonable options for you and plausible possibilities, you're probably kidding yourself on one score or another. Generally if you'd even be seriously looked at for a TT job at University X, then when the postdoc committee looks at your file, they'll say "We could hire this person for a TT job. There's no way that a postdoc at University X is the best job they'll get." and they probably won't offer you the position. But unless you feel confident about which way it is, I don't see any problem with applying to both.

EDIT: One thing I'll add, which is mathematics specific: if you're applying MathJobs, the visual difference between applications for the different jobs is negligible. There is a column where you can see what positions the applicant applied for, but it's not very noticeable. So, it's quite possible the committee for one job won't even notice you applied for the other (I have seen this happen).

As a recruiter, I don't see much of Chris's point. And, as a candidate (some years ago), I followed pretty much your option B (apply to both). Here's the reasoning:

Times are though, and the academic job market is very competitive. Anyone who expects you to apply only to a single position (“applying to more that different positions shows that you don't know what you want”, or “you don't believe in your chances”) is delusional to some extent. You will apply to backup post-doc positions, and I don't think you benefit by excluding the place where you also apply for faculty position.

In fact, you can actually turn that into a pretty positive argument: “I love your institution/department, this is really my dream place to work at, and if I don't get on faculty this year I'm willing to take a post-doc, know you all better and try again next year (if circumstances permit).”

I'll add that this is actually commonly done in France, UK and some other European countries: once you've traveled a bit, find the place where you want to settle, apply there, and if you don't get the job on the first try, get a post-doc there and try again later.

One thing to be aware of that I don't think others have mentioned is timing. In my field (theoretical particle physics in the US), postdoc hiring decisions are all made in December and January, while interviews for faculty jobs usually take place in February, March, and possibly April. Postdocs who are on the last year of their current job have to apply for and accept a postdoc position, even if they are hoping to get a tenure-track offer, simply because of the timeline. If they haven't accepted a postdoc job in January, and they don't manage to get a tenure-track offer that spring, they have no job at all. Because of this, everyone expects that people will apply for both postdocs and faculty positions simultaneously.

The timing of these things will potentially vary depending on your field, so this consideration might or might not be relevant. But it's something to think about.