The title pretty much says it all. Must I forgo all other masks if I want to find a level's secrets? I will cope with scouring for secrets with greater effort, but I'd hate not to be able to find them at all.
A bit of extra effort is all you will need to find the level's secrets if you choose to not equip the Rasmus mask. Wearing Rasmus makes the clues flash and bob up and down so they are easier to spot; however, not wearing Rasmus does not prevent you from seeing and grabbing them. To wit:
Rasmus mask equipped; it's bobbing up and down and flashing, you can't miss it if you're looking.
Note that Rasmus does not make any other secrets in the game easier to spot - there are a couple of masks hidden in levels, and it doesn't do anything to point them out to you. However, save one they are pretty hard to miss - just make sure you interact with things that look interesting.