It's been a long time since my metallurgy courses, but in general there are three major ways that properties are altered at the mesoscale due to alloying:

1) a different solid phase of the dominant metal is formed due to the solid solution of the minor alloy components

2) covalently bonded compounds are formed between the base metal and minor components ("intermetallics")

3) the microstructure is changed (think e.g. alternating layers of intermetallics and metal grains)

Because the thermodynamic driving forces are basically identical regardless of the solid phase, #1 is probably not helpful here (also, it's flammable in liquid form -- I checked). But the intermetallics may not be flammable. So then your fire resistance comes from a combination of #2 and #3: if you form a lamellar structure of the non-flammable intermetallics separating the regions of the flammable majority phase, then that may produce macroscopic fire resistance.

I can't comment on how well it works, of course. But if you have some "secret sauce", maybe a sintering process or special heat treatment, you might be able to manipulate the lamellar structure in a desirable way and end up with a very fire-resistant alloy. The catch is that this might be pretty expensive.

ETA: because this fire resistance is dependent on the microstructure and presence of intermetallics, if the alloy is heated back to a temperature where those intermetallics dissolve into solid solution (or, obviously, if it melts), then it's going to be flammable again. So my educated guess is that while you can't actually light these special alloys on fire, if you were to throw some into an ongoing inferno, it would heat up and then combust. So in an otherwise flammable environment cough hydrogen airship cough, maybe not the greatest idea.