I'd like to see "interruptions" happen only when the DoT damage (or incoming arrow damage) is a considerable part of your health. Or maybe with an appropriate check of some kind. If a 1st level character with 30 hp gets hit for 6 points of damage, I can see why that might interrupt them from opening a door or pulling a lever (I think they would still be able to, but at least it sort of makes sense if they can't). When an epic level character with 1200 hp gets hit for 6 points of damage, I doubt they'd even notice, must less be so distracted by it that they could no longer do mundane tasks such as opening a door or pulling a lever.
Perhaps if the damage is more than 25% of your total remaining health, you have to pass a CON check or make a FORT save or something like that.