Fuels and camp stoves on a plane
Fuel cannot travel as passenger baggage
For a departure from the US, do not pack gasoline, butane or propane in carry-on or checked baggage. This includes camping cartridges and lighter refill cans, even if the container is small or partly used. The ordinary liquids size rule does not make flammable fuel permissible. TSA also lists gas torches as prohibited in both bags.
A stove needs to be free of fuel and vapor
TSA permits a camping stove in carry-on or checked baggage only when all fuel has been removed and no vapor or residue remains. FAA notes that an airline may still refuse used equipment after cleaning. Check the carrier's policy before bringing a stove that has previously held fuel. A fuel bottle or cartridge must be assessed separately from the stove body.
Check each ignition item separately
An ordinary lighter or safety matches follow different rules from a gas torch or spare fuel. See the lighter and match guide before adding either to your kit. Confirm the conditions for every part at departure and on any later flight with another airline.
Sources: TSA: item list and FAA PackSafe: outdoor equipment.