Actually the heat alone is enough to produce colors, depending on the material. The energy in chemical reactions is due to a change in the energy of the electrons, which can have different levels of energy, depending on the atoms / molecules invovled. In the case of fire, the combining of oxygen atoms with other atoms (or molecules) results in the electrons releasing energy and ending up in a lower energy state. This release of energy is generally in the form of heat (infrared light waves). If there's enough heat generated, it also creates higher frequency waves that can be seen.
With an electric oven, you can still get heat and light without any fire, just the increase in energy of the molecules generates the heat, and the temperature determines the color viewed.
If you've ever had a chemical cold pack, this is a chemical reaction that consumes heat. After breaking the back and letting the chemicals combine, the pack gets cold because the chemical reaction results in absortion of energy, with the electrons going into a higher engergy state.