In general, you have cause to investigate something whenever you think that the chance of something being an illusion is significantly higher than the chance of it being real. Illusionists might be rare, but they do exist within the game world; maybe you ballpark a one-in-a-million chance that any given object of interest you see is an illusion. In that case, you should try to disbelieve anything you see that is substantially less likely than one-in-a-million. If you have reason to suspect that an illusionist is nearby, then that increases the base odds of something being an illusion, so it lowers the threshold of skepticism before deciding that something warrants investigation.If you've ever played D&D back in the day, it became something of a habit for players to try and disbelieve anything that seemed too implausible. If you're wandering through a volcano and you encounter a snowman, then you know something weird is going on, so you will probably stop and investigate to find out whether it's an illusion or magical item or some sort of trap. If the only bridge across a moat to an enemy castle is suspiciously unguarded, then you know something weird is going on, so you will probably stop and investigate to find out whether it's an illusion or some sort of trap. Around these parts, that investigation usually comes in the form of shooting it to see whether it reacts or if the arrow simply passes through.
Depending on the player and their base level of skepticism - and what sorts of DMs they've been subject to in the past - they may opt to investigate everything. If you've ever seen a player stop and poke everything in the dungeon with a ten-foot pole before getting close to it, then that's probably a player who estimates a fairly significant chance that something isn't what it seems, even when they have no obvious reason to suspect that thing in particular; they wouldn't waste their own time if they didn't expect it to yield results.
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