I just got a job offer in the Netherlands, meaning that I need to get a TEV (entry visa), and an MVV afterwards (residence authorization). I started the process for the TEV with a passport number that I currently don't have with me anymore, meaning that I just lost it. It didn't have any relevant visas. Could I just get a new passport and update the application? Or do I need to start a new application from scratch using the new passport?
First, as @EugenMartynov suggests, it never hurts to ask. Having said that, and without actual experience with your situation:
Nobody assumes 1:1 relation between people and passports
Remember that a passport is merely an identification document. You are not your passport, and the passport attests to a proper unique identifier combination (e.g. name + birth date + birth place). So it does not seem reasonable that you should not be recognized as the same person because a different document attests to it.
My slightly-relevant experience with this situation is not Netherlands-related, but I once started some formal procedure vis-a-vis a consulate of Romania, being an Israeli citizen and using my passport. At some point I lost my passport, got it replaced, and simply used the new passport number in subsequent steps of this procedure. No questions were asked.
Use corroborating documents for the passport change
When continuing the process using your new process, always have a copy of your old passport handy (e.g. photocopied or scanned and printed). That will help if you do encounter trouble. Some states may also issue confirmation that a passport has been invalidated or canceled; if that's the case in the state you live in, have that handy as well.