I believe that interacting with involves tactile sensation in some way. I could be mistaken, but I think if I make an illusion of a giant praying mantis at the end of a hall, they only get the save if they attack it (or if my illusion attacks them) or if they otherwise physically move to a place where they'd be in physical contact.

I might alter this a bit in some other ways. If I make the illusion of a fire elemental I might allow a save if a player was close enough to feel the heat of such a creature pouring off of it, or if I make the illusion of a tray of limburger cheese a player might get a save if they get close enough to it to smell it. Those are sensory things that have an "effect" on the player.

This is why illusions to hide doors (or gaping holes in the floors) are extra tricky.