When the electrons jump up to a higher state, they almost immediately fall back down. At the higher state they have a higher energy which makes them not as stable, so as soon as whatever influenced the electrons to move upward has finished acting upon them, the electron jumps back down and emits its excess energy as light. In any visible sample, there are so many electrons undergoing this "jump" that the color becomes constant.
Remember, both heat and light basically occur as a series of waves, so the influence of it is not constant. Light occurs as the commonly described electromagnetic radiation, or a photon, and heat is a result of infrared radiation. All of these are not 100% continuous events. If you look at light as a particle, the particle hits the electron, excites it, and then is gone. Once it's gone, the electron falls back down. If you look at it as a wave, the peak of the wave hits the electron, making it jump up, then the downside of the wave hits and the electron falls back down. This is repeated ad infinitum. So in essence, the color you see is a result of continuous excitations and relaxations.