So here's a visual of the optimum way of doing this, simplified (the values radiating from the center of each fire would really be something like 15, 14.9, 14.8, 14.7... to zero)
One way to go at this is from the other direction. When a fire is placed, there's no way I can pre-fill every possible coordinate that the fire could heat. But maybe if I simplify it...
Look at the world as several layers of grids. The largest would have few cells (12x12?). The smallest would have a huge number, and get down to very local info. Have limited number of layers (5?10?)
When a fire is placed, is would be sized to the largest possible layer (maybe if it contains a % of that layer. If no %, then go the smallest). It would ONLY be placed on one grid at a time.
When a temperature calc is requested, the system would find which grid the user is on for each layer, and sum the temps in those grids (so 5 layers would return a max of 5 cells to calc - for example: world wide, regional, state, city, and local. If no value had been placed in the local cell, then it wouldn't get a value for that cell (or it would be zero).
So ANY fire placed, no matter how large or small, would have a calc on a small set of cells. In the DB, only those cells that had been effected would actually have a value (so in the above, layer 1 would have 12 cells populated, layer 2 would have 121, and layer 3 would have 63.
Fires would normally be placed rarely, while requests for temps would be done requested very often - so this might work. The down side is that there's not a easy way to decrease the temp of a single fire over time (which I was planning on doing). Might be able to force a periodic refresh, but that would be annoying.
Above is for biggest fire, which would be world wide - The center is in on two of those big cells, so effect is highest there. Surrounding would be less.
Above is for a smaller fire - it's on a mid-level layer.
The last is on a small-level layer, with only a very local effect.
Didn't show it here, but I think as it goes down, it would effect more of the (smaller) cells.