In order to acquire heat from objects (rather than from being attacked), the agents take a picture of the incriminating object. They certainly aren't using DSLRs, as they need to get within one tile of an object to trigger the animation. This means that they can't gather heat from objects behind doors.
The startled "oh my goodness what's that? It appears to be nefarious!" animation can be triggered from a few tiles away but only within the same room and where the agent has a viable path to get to it, so again not through field barrier doors.
The viable path requirement means that the agent must be able to walk to within 1 tile without their path being obstructed in order to trigger the "gasp" animation and consequent heat production - for example during the base move my Ark of the Covenant Uber Loot became stuck in the corner of a strong room behind piles of impassable gold. Despite several "exceptional" agents appearing to have a line of sight to the loot, they ignored it entirely as they had no way to get within one tile of it.
Therefore your ill gotten loot is entirely safe behind a field barrier door as long as the agents stay on the other side!