This could be a long post, but this is something I actually know a bit about...

Various posters have already noted that tidal effects fall off more rapidly than the inverse square law of gravity.

In fact, tides are due to gravity gradients, and therefore fall off as the inverse cube. This is why the tidal effect at the Earth of the small Moon is more than that of the large Sun, and yet the Earth orbits the Sun, not the Moon.

Tides on the Earth cause a distortion of the figure of the Earth. This solid-body tide can be measured directly with long-baseline strain guages emplaced in tunnels etc, or by laser-ranging between mountains, or even the towers of long suspension bridges! It can also be measured indirectly. If it did not occur, ocean tides would be higher than they in fact are, since the water would rise past a static coastline, not a rising one. (The full calculation of tide heights is very complex, involving resonances in shallow water and amplification effects. The true oceanic tide is ~1m in amplitude.

The tidal bulge is not directly in line with the Earth-Moon vector. It lags slightly, due to viscosity, therefore the tidal bulge is not directly under the Moon. (Observing this is difficult, because it is a small offset, and the Solar tide complicates things). Nonetheless there is a phase lag between the bulge and the gravity vector.

This phase lag allows the bulge to act in turn (by normal gravity) on the Moon, dragging it in its orbit. If the Moon were in a lower-than-geostationary orbit, or in a retrograde orbit, the drag would retard its orbit and it would spiral down until impact (or until it reached the Roche limit, where tides become stronger than the gravity holding it together). This would not be a good thing.

Mars' Moon Diemos is in such a low orbit that it's lifetime is only another ~50 Million years from now.

Our Moon is beyond geostationary and prograde, so it is being accellerated in its orbit and is retreating from the Earth at a few cm/yr (3.82+-0.07) - similar to the rate of plate motion. At the same time, this tidal drag robs energy from the Earth's spin and slows it down, increasing the lengyh of day by milliseconds per millenium.

We can extrapolate this forward and backwards in time, and measure past day, month, and year ratios from a variety of special geologic sediments. Growth bands on corals, tidal sediments, and glacial varves are all useful for parts of this measurement. Tree rings can also be used if well preserved and fast enough growth rates, but fossil trees don't go far enough back to be real useful.

One of the best datum points comes from fossil Permian corals which show that the day was ~16 hours, the month ~25 modern days or ~33 Permian ones. (Unfortunately, my referenc ebook on this is in the Uni and I'm home. These figures are broadly OK, but don't quote them as definitive).

A good online reference is here.

When we try to fit together all the observations of observed oceanic and solid body tides, and rate of change of the Moon's orbit, we get a pretty good balance when we allow for an internal tidal dissipation *much* smaller than that measured in the Ocean tides.

From this we can measure the mean viscosity of the Earth's mantle, and the rate of tidal heating. The former is large (approximately equal to mild steel), the latter rather small, compared to radiogenic heat.

So, to wind up a long and rambling post, there is little direct thermal effect on the interior of the earth, due to it having a Moon and tides.

_____________________________________

In the past, the Moon was closer and the tides larger (10 times closer, 1000 times larger when the Moon was first formed - at 4.5 Ga). Then, the effects were much more (1 million times more heat).

There would have been significant effects then, even compared to the higher rate of radiogenic heat in the past (up to 10 times higher than today).

Imagine a Moon with an apparent diameter of 5 degrees, looming overhead while day lengths were ~3 hours. Twice a day, a 1km-high ocean tide is raised and rushes across the land, scouring and eroding it.

This time didn't last long - the Moon was dragged away by tidal forces at a rate ~1/2 million times faster than today - 20 km/yr. So within 1000 years, things would have calmed down by about a factor of 3.

[Anyway, this was so soon after the Moon-forming impact that the Moon was still assembling the last pieces of debris, crashing into its Magma ocean, and on Earth, the surface was similarly hellish and there probably wasn't even an ocean - just a thick steamy atmosphere glowing dull red.] - but this is all a different story...