In Fahrenheit 451, what does Beatty tell Montag to burn his house with?
The answer to your question is simple: Beatty tells Montag to burn his house with a flamethrower. It says as much in the following quotation:
I want you to do this job ... not with kerosene and a match, but piecework, with a flamethrower.
This happens when Beatty and Montag go to Montag's house in Fahrenheit 451. Beatty knows that Montag has taken a liking to reading and to books. This quotation above is a demand by Beatty that Montag give up that curiosity.
There must be something in books, something we can’t imagine, to make a woman stay in a burning house; there must be something there. You don’t stay for nothing.
The irony is that Montag is a firefighter whose job it is to burn the books that he is interested in! And why does Beatty suggest a flamethrower as opposed to a simple match? The answer lies in the degree of intensity of the action. Beatty desires Montag to fully reject his love of books. Therefore, Montag, can't just light a match and walk away. Instead, Montag must be the instrument of destruction by engaging the flamethrower and watching the books burn.
After Mildred turns him in, Montag and Beatty go to Montag's house. Beatty, knowing that Montag has been infected with curiosity about books, demands that he burn the house himself:
"I want you to do this job all by your lonesome, Montag. Not with kerosene and a match, but piecework, with a flamethrower. Your house, your clean-up."
(Bradbury, Fahrenheit 451, Google Books)
This is meant to force Montag to reject books and his own interest, to return to his work by ridding himself of the very thing that he is supposed to stand against. By using a flamethrower, Montag will be confronted with everything he had, and everything he threw away by keeping books; the personal connection will be more powerful than simply dousing the house in kerosene and lighting it with a match. After he is finished, he may be allowed to reenter society, but Montag's refusal to accept his role helps him to escape after burning the house.