When methane burns, energy is released as heat and light. Does the energy released weigh anything?
Yes, the weight of light/heat is very small but nonzero. The weight of something is, by definition, is the magnitutude of the gravitational force that acts on it.
Now, according to the theory of General Relativity, gravity couples to a quantity known as the "energy momentum tensor", which is pretty much the same as the energy content.
If an amount of energy E is released and you could somehow capture it in an empty box, the mass of the box will increase by an amount E/c^2 (c is speed of light). The weight of the box being proportional to the mass will thus also increase.
Note that the mass of a single particle of light (so-called photon) is exactly zero and not equal to its energy divided by c^2. This is because "mass" by definition means "rest mass" (contrary to what you can read in many popular and high school physics books)
Nevertheless, gravity will couple to the photon with a strength proportional to its energy, not its mass.
If you have a box filled with photons that move in random directions, then the total momentum of that system is zero and then the energy of the box divided by c^2 equals the mass.