Actually, KD, the PC rolling on the ground trying to put the illusory fire out is showing me that he's NOT got any doubt it's real fire. So, until he wakes up from the 0HP, he THINKS he is burning to death.
Yes, initially. But as part of that process, the flames are not going out and he is interacting with the illusion.
This also results in a double penalty for the spell that shouldn't exist.
You are rolling on the ground trying to put out the fire. That takes up an action (or minimally Movement).
And, you are taking auto-damage from the fire.
When compared to something like Hold Person which gets a save every round and the PC loses his action and he grants advantaged attacks, at least the PC gets a save every round. Getting two saves total (the normal one when the spell is cast, and an Int check later on) should mean that the PC is not also wasting actions every round and taking damage without a to hit or a save every round. IMO.
This sounds extremely potent and abusive if interacting with the illusion (and using up an action to do so) does not result in the Int check.
Additionally, your comment here illustrates the OP's point. If the character really thinks that he is on fire, he would never take the time out to "use an action to make an Int check". He would try other worthless actions to accomplish the unattainable goal.
But from the PC's perspective, what is an "examine the phantasm with an Intelligence [Investigation] check against your spell save DC" action? Does one stand there an ponder it like Sherlock Holmes? Meh. I prefer that the character interact with the illusion to "examine it". Patting down the fire with one's hands, rolling on the ground, all of these (in my mind) mean that he is closely examining the illusion as opposed to ignoring it.
Another aspect of this is that this is only spell (TMK) that uses this mechanic. If one's adjudication typically minimizes the mechanic from being used (by forcing the player or DM to explicitly declare an Int check action by the PC or NPC), then why have the mechanic in the spell? So that players can rarely guess the right answer when an NPC casts this spell on them?