[derefr]

I've always wondered if there was something you really could do to neutralize a reaction that "really wants to happen" like that. Presuming you're in a well-equipped chem lab and have access to all sorts of non-mundane substances... maybe you could pour liquid helium on the fire? (It'd splatter from the Leidenfrost effect, but the flying Helium droplets wouldn't harm anything in any chemical sense, "merely" a physical "being very cold" sense. So, suit up against frostburn, and move away any glassware to ensure it isn't shattered by the opposite of unequal heating.)

[kortex]

ClF3 will react until there is nothing left to fluorinate. In practice, this means it'll react with most neutral elements, oxides, and chlorides. Certain oxides might act to poison the reaction, slowing it down by absorbing free radicals. Mainly, you wouldn't even want to be close enough to such a reaction to be able to douse it.

[slavik81]

I don't know about chlorine trifloride, but that is one of the strategies used for reactive substances, like molten sodium: https://youtu.be/rAYW9n8i-C4?t=2m