Indian lore says to kill burdocks after the second full moon in the spring. Supposedly that prevents them from coming back. It must have to do with the weather patterns and the timing of the release of the spores/seeds.
I generally cut them down, then compost them. Don't leave them in the field and don't burn them. If you don't have a hot, active compost pile, put them in garbage bags. Then you can either throw them out in the trash, or after 4 - 6 weeks of being overheated, you can burn them or put them in a regular compost pile. They will be soft and mushy, so not so easy to burn.