What has been the influence of the moon on Earth's temperature?
The complications of calculating the moon's contributions to earth's heat budget are touched on in Emergence of a Habitable Planet (2007) by Zahnle, et.al. in Space Science Reviews, a paper trying to determine   when in earth's history the planet's surface would become cool enough to support life. The portion of the paper that relates to tidal heating suggests that the early moon contributed substantially to the interior heat of the earth system for about 1.2 million years, but as the moon's orbit widened, the earth cooled and solidified, and water condensed into oceans, the moon's contributions gradually became minimal.


  Viscous damping of tidal motions generates heat. Therefore tidal
  heating occurs most strongly in materials that are solid but close to
  melting. This introduces the possibility of a governing feedback that
  works through the dependence of viscosity on temperature. If tidal
  dissipation exceeds what the atmosphere can radiate, the excess heat
  raises the temperature, which lowers the viscosity, which in turn
  lowers the rate of tidal dissipation. This looks like a stable
  feedback. It follows that, while tidal dissipation was important, the
  base of the mantle was solid but the rest of it was fluid, and tidal
  heating generated almost all of the thermal energy radiated to space.
  In the limit of an asymptotically thick steam atmosphere, tidal
  dissipation would have been regulated to generating heat at the
  runaway greenhouse limit of ∼140 W/m2.


In a caption for a figure the authors also note:


  Tidal heating plays an important role in prolonging the magma ocean.
  Tidal forcing wanes as the Moon evolves away from the Earth.
  Thereafter heat flow is controlled by convection of the solid mantle.
  By 4.4 Ga the global average heat flow would have been ∼0.5 W/m2.
  Later in the Hadean typical heat flows would have been 0.2–0.3 W/m2,
  not enormously larger than what they are now. For comparison heat flow
  today is 0.065 W/m2 through the continents and 0.1 W/m2 through the
  ocean crust.