For my first example, let's say that a creature has contracted Filth Fever, and subsequently gains immunity to disease. The source could be anything, a Periapt of Health, or perhaps they gained their third level of Paladin, granting them the Divine Health ability.
What happens to the disease?
Is the disease removed completely and instantly? (this seems highly abuseable)
Do they retain the disease, but its effects are suspended until they no longer benefit from the immunity, ie taking off the Periapt, or falling and losing their Paladin class features?
Do they continue to make saves against the disease, but without suffering the negatives of failed saves, until eventually they save against the disease and shake it off? (This one makes the most sense to me)
Do they continue to make saves against the disease and suffer its negative effects until they are cured?
Second example: Fear effects.
Lets say a Paladin and Antipaladin are in a throw-down. The Antipaladin uses intimidate to demoralize the Paladin, which, due to his Aura of Cowardice ability, allows him to succeed and makes the Paladin shaken for 4 rounds. The Paladin backs up to outside the aura, regaining his immunity, then walks back in. Is he still shaken, or is he fine now?