FASA Games, Inc.

It's very dependent on style (the blessing and curse of illusions being treated as its own thing), but for the most part:

regular Disbelief: you can always make this, but it's a Standard Action and costs the Strain
non-regular Disbelief: is sort of a resistance test and is prompted, such as in Displace Image, and this is rare.
Sensing: any test made against an illusion automatically senses

Disbelief is much like with WIL Tests vs. fear in regards to when it happens and if it's a Standard Action. If it specifically mentions/prompts it, it's usually free and automatic.

Now, some GMs require some sort of justification for the regular Disbelief. It's a decision based on principle, but the rules don't require it.
The more ambiguous thing is situations where it's in doubt whether or not you should get a Sensing, or whether or not something you do counts as a test against the illusion and hence is a Sensing Test. You're basically looking around all the time, so some would ague you're entitled to constant PER tests, and as such have a constant chance to Disbelieve. By the same reasoning, however, you should be making DEX tests constantly or you'll trip.