You should contact your potential references now. You need their permission to list them as references before you submit any applications.
Does the application system request their reference letter as soon as I apply?
Every department is different. Some do, some don't.
Does every application requires a reference letter, or do my references only get asked by the system if the application seems promising?
Every department is different. Some ask for letters for every candidate; some only request letters for candidates that satisfy minimal qualifications (like having a PhD); some only request letters for candidates that pass a more rigorous initial screening; some may even wait until after interviews. My department automatically asks for letters for all minimally-qualified untenured candidates, but we only request letters for tenured applicants when we decide we want to interview them.
Or should I email each of my references directly after I apply to request their recommendation letter, and if so, where do I tell them to send it?
Every department is different. Some will tell you to ask your references to send letters when you apply. Most departments will ask for contact information and request letters directly from your references.
In any case, each department will provide their own instructions. Follow them.
I've recommended the following protocol to my students for several years. When you decide to apply, you create a shared spreadsheet (on Google Docs, for example) with one row for each application target, and columns listing the university, department, a link to the ad, instructions for submitting letters (as supplied by the department), submission deadlines, and each of your references. Ask each of your references to indicate in their column of the spreadsheet (1) when anyone asks them for a letter, and (2) when they have actually submitted a letter. That way, you can keep track of the status of your applications and letters, your references have a unified record of where and when they need to send letters, and your references can be properly (but not too publicly) embarrassed if miss a deadline.
