I am currently applying for post-doctoral and asst. professorship positions. One of my letter writer is most likely to submit the letter a week late. Some of the positions I am applying to clearly state that "applications submitted after the deadline cannot be considered". Couple of people (my fellow grad students/post docs) whom I talked to, seemed to indicate that the reference letters can reach late. However, I would appreciate if someone (preferably one who has been part of the search committee and/or has reviewed applications for such positions before) can throw some light on this.
Unless the search committee is swamped and is looking for reasons to reject a candidate, a late letter will not tank the application. But here's what can happen: letter requests are sent out for candidate A and B. B's letters are delayed. In the meantime, A's letters come back and they are quite good. The committee starts going ahead with interviews for A. Then, for other reasons, the committee feels that A might suffice to cover the area that A and B both overlap in, and they might move on to other candidates.
This would only happen if the letter is REALLY late (multiple weeks). And if candidate B has any level of interest, there's always someone who'll chase down the letter writer.
So overall I'd say not to worry greatly, but make sure the letter isn't more than a week or so late. Also it wouldn't hurt to check with the committee on the "hardness" of the deadline.
The policies for late letters may vary, depending on the sort of position. My impression is that for most universities (certainly the ones I have been at), slightly late letters are fine as long as they arrive before they are needed. I.e., your application may suffer if people read it before the letter arrives, but they are unlikely to read it on the day of the deadline. This is in agreement with the other answers here.
However, there may be certain cases where a single late letter will lead to absolute rejection. For example, government funding agencies sometimes have very strict rules about deadlines, in order to guarantee a sort of formal fairness. They may say that nothing will be accepted after the deadline and no incomplete applications will be considered. (I don't know how common this is, but the NSF graduate research fellowships work this way.) If you are applying for a postdoctoral fellowship from a funding agency, then I strongly recommend asking what their policy for late letters is.
The pure and simple answer to your question is answered by asking a question to the committee: when will you start reviewing the applications? Usually it's not the day after the deadline, likely a few days later, and maybe a week later.
The same happens for PhD applications, some departments don't start until two weeks after the deadline. If all your material is in then I am sure they will look at your application regardless of one missing letter.
Just call the department and ask when they start reviewing.
Different universities take different approaches to letters of recommendation. Some departments request letters of recommendation for every candidate; others only request letters when they're highly interested.
In general, however, the difference here is that the letters of recommendation are not normally considered as part of the application itself. Thus, a little leeway is certainly possible, particularly when you'll have "N – 1" letters of recommendation in on time. But again, as others have mentioned, the best advice is: when in doubt, ask.
Upvoting and reiterating @Suresh's comment: a letter just a little late does not seriously disqualify you (except, as Anon Math'n notes, perhaps for government agencies, NSF grants...), but may allow other people into the pipeline before you.
The strategy on job offers is typically to make more offers than a dept expects to have accepted, and to have quite a few of the first offers rejected, because often the most-attractive [sic] candidates have several offers and cannot take them all. Thus, if the delay in your letters is a week or two, it will probably have no impact.
However, the potential for trouble grows sharply as the delay increases, insofar as scheduling for interviews is made as early as possible... and unexpectedly high acceptance rate may use of all the offers unexpectedly early. (Not bad from a dept's viewpoint, but bad from not-yet-interviewed applicants' viewpoints, since, in effect, they didn't have the same chance... but through the early error of incompleteness in their file.)