Speaking as a layman, I understand a laser works by exciting an electron so it returns to rest by emitting a photon. Is it feasible to 'damp' the energy level/s of an electron so it absorbs energy instead?
Lasing is a bit more complicated than you describe.
If you take a gas and excite the atoms in it by whatever method is convenient, you'll find that more atoms are in the ground state than in the excited state. The populations will generally follow the Boltzmann distribution so higher energy states have lower populations. In this state a gas won't lase.
To make a gas lase you need to create a population inversion i.e. have more atoms in an excited state than in a ground state. It's only when this is the case that you get lasing. This is hard to do, and the various different types of laser use a range of methods to achieve the population inversion.
Anyhow, to get back to your question, a normal gas (i.e. no population inversion) will absorb light then re-emit it in random directions. This is almost what you're asking, but the light is only scattered and not absorbed. To get light absorbtion you need some mechanism for quenching the excited state without allowing it to emit a photon. Quenching tends to be most pronounced in liquids and solids where the energy can be dissipated as lattice vibrations, however I think quenching can be caused by collisions in gases. If so any old gas will absorb light as you want and you don't need to do anything special.
Yes, the inverse of a laser does exist. It's called an "anti-laser" or a "Coherent Perfect Absorber" [1, 2]. You basically use interference to "force" a material to absorb incoming light. Instead of using population-inverted electrons to create a coherent beam of light you are using a coherent beam of light to invert a population of electrons (and can extract the electrical current).
 Feng, Simin, and Klaus Halterman. "Coherent perfect absorption in epsilon-near-zero metamaterials." Physical Review B 86, no. 16 (2012): 165103.