If it is burning, how do you put it out?
You are confusing fluorine with fluoride. A fluoride will not burn because it has already reached a state with a potential lower than that it would have with water or air.
With that said most every LFTR design I've seen does have fluorine as a gas at some point in the process but that is in the chemical processing of the fuel while outside the reactor. There is little to no fluorine in the reactor vessel.
There would not be a fire because the stuff in a LFTR does not burn. If there were things burning then the answer is to use water.
both uranium and fluorine are very toxic elements.
Uranium tetrafluoride is an insoluble salt, no more toxic than sand. Saying uranium and fluorine are very toxic is like saying sodium and chlorine are very toxic. Sodium and chlorine alone are very bad but combined they create a substance vital to life. I suppose you think we should ban the use of table salt because of the toxic materials it is made of.
What's the worst case for LFTR?
The worst case is you douse it with water for hours, maybe days, until it cools off. After it's cool you send in people with jackhammers and tractors to haul away the pieces for recycling. The mangled mess would no doubt contain radioactive material but since fission would have been stopped for days at this point the pile of scrap would be about as radioactive as a typical granite counter top. The workers would have to wear protective gear for the dust because heavy metal poisoning is a risk, just like for people that mine for gold or coal.
Perhaps I am mistaken, perhaps I exaggerated a bit, but regardless a LFTR simply cannot burn or react with water like you describe.