No, it's not needed. You can do this because you want to (for whatever reasons, even if only for aesthetics) but it's absolutely not a requirement.
Waterproofing seems to be legit case.
Only where you expect the piece to come into regular contact with water.
Given how long the material has been with us there have to be hundreds of millions of furniture items with unfinished edges on some of the particleboad/chipboard components and in general they hold up well. There's no problem from the water vapour in the air (humidity) and even the steamy environment in a bathroom may pose no threat. So sometimes its sensitivity to water is slightly exaggerated.
Some say the ends should be covered to stop formaldehyde fumes from spreading, is it true?
Probably not. The main problem with this line of reasoning is that if the plastic/resin veneer coating doesn't stop the egress of formaldehyde (if present in the board) what chance does filler have?