Is it better to indicate 3 interests rather than one? Does it affect who looks at my file or affect my chances of acceptance in any way? For example, suppose I'm mainly interested in combinatorics but also am interested in algebraic topology. If I just put down combinatorics as my primary interest and nobody in the department does combinatorics, will that make the file worse than if I had put down 2 interests?
When I, as a faculty member, go to MathJobs to evaluate postdoc applications, the first thing I do is to click a link that sorts the applications by research area. That way, I can efficiently go through the applications in my field without having to scroll through the whole database. As far as I can tell, the sorting is done on the basis of the applicant's primary interest. So the answer to your title is yes, your choice of primary interest definitely matters.
If the job has a heavy teaching component then a variety of interests is probably valued. For a primarily research job, however, having interests that are aligned in some way with current faculty will be helpful as it adds synergy. In such a case, specialization would likely be valued over generalization.
Do your homework and find out what are the needs of the department you would be joining. I assume that for a post-doc, research is the more important thing, but it should also be compatible with current faculty interests.
However, if you are in a particularly hot/new area, then being unique in a department might be an advantage, especially if other members of the faculty were interested in moving in your direction. That would be more likely in a regular position, of course.
For the MathJobs database, you have a delicate balance, depending on the job you seek. If you want a research position, don't give the impression that you are a dabbler in many fields, but that you are focused on your primary area. You can also, perhaps, say that you have secondary interests, especially if the support your main focus in some way.
But, be honest in all things. It is a mistake to "oversell" yourself. Be sure that you are competent in any field you claim as your own.