As one of the six noble gases, neon is very inert and nonflammable. In can be made to form a compound with fluorine under certain lab conditions, but otherwise it typically does not react with other elements.
The glow from neon signs is caused by electricity, not combustion. Electrodes on either end of a glass tube alternate between positive and negative charges, which causes electrons to crash into the neon molecules and dislodge more electrons. As the neon molecules grab more electrons to get back to their normal state, they give off light.
The noble gases are nonreactive because their outer shell of electrons is full, which means they have no incentive to trade electrons with other elements. The other five noble gases are helium, argon, krypton, xenon and radon.