I don't actually know how much influence tidal forces have on the Earth's interior temperature. But here are some thoughts to consider.

The Moon very small compared to Earth and much less massive. To get a sense for that, compare the Moon's entire radius (~1740km) to just the radius of Earth's core alone (~3480km). Just our core alone is twice the radius of the whole Moon.

The Moon is made mostly of rock, like our mantle. Rock is very not-dense compared to metal. It's so not dense that the change in density from the deepest part of the mantle to the shallowest part of the outer core is MORE than the change in density from our atmosphere to the crust.

And then our mantle (rock) adds on another 84% of Earth's volume.

So while there is going to be some amount of tidal heating experienced by Earth on account of the Moon, I suspect it's likely not significant. Earth is just SO MUCH more massive than the Moon. (But hey, I could end up being wrong!)

So there's an interesting story here, but we need to go on a bit of a long walk involving lava lamps, a magnetic-field paradox, and iron snow, to see the whole picture.

Currently, we believe that almost all of the planet was molten liquid when the Earth finished forming, due to the intense heat from all the billions of collisions that make a planet. Since then, Earth has been slowly cooling by conducting its heat up through the mantle and then radiating that heat out into space. Because space is cold and the centre of Earth is hot, we might expect that the Earth would "freeze" (solidify) from the outside inward.

More rock means better insulation, and so a larger planet can insulate its liquid core better. This is why small bodies like the Moon (and Mars?) don't still have a magnetic field--their cores cooled quicker due to having less insulating rock atop their cores. Earth is the biggest rocky planet around the Sun, and has a very well-insulated core.

But there's a big problem with this, and it has to do with the question of exactly how insulating is that rock? The classic theory about our magnetic field is that there is convection in our liquid outer core. Convection is what makes a lava lamp: the hot lightbulb at the bottom of the lamp heats the wax, and as the wax heats up it expands, and as it expands it becomes less dense than the water, and so it floats to the top like a lifejacket. Colder wax at the top contracts, becomes more dense than water, and sinks.

If you somehow had a liquid-iron lava lamp, this convection would also create electrical currents and a magnetic field (because physics, I dunno). Our liquid-iron outer core has long been thought to work like an iron lava lamp: the deepest part of the outer core gets heated by the (VERY HOT) inner core. That makes the liquid iron expand and rise up. At the top, it conducts heat away into the mantle. The liquid metal now contracts a bit and sinks again. These circular currents create the Earth's powerful magnetic field.

Except, they don't. A few studies about a decade ago (ish) started showing that the mantle (which is solid rock) conducts heat too well. Heat is leaving the core faster than we thought. Scientists know how old the Earth is, so, when they compared the measured rate of heat leaving against the age of the Earth, they discovered something shocking: the core is now too cold to run the heat engine to make our magnetic field!

This would be apocalyptic, except for the fact that we very much do still have a magnetic field. It doesn't make sense. So how do we explain that? Something else clearly must be making the liquid iron circulate.

The answer to this lies in the answer to a question I've had since grade school: WHY is there a solid inner core and a liquid outer core?

Remember: scientists have long agreed that probably almost all of Earth was liquid at the start. The inner core didn't always exist. As the convection of liquid iron carries heat out to the mantle, the whole core itself is cooling over geologic time. Eventually, some of that liquid gets cold enough to crystallize. Particles of solid iron grow and (being denser than liquid) sink as flakes of iron snow downward. This slowly grows to form an inner core of the Earth!

Finally: even if we don't have enough heat to drive circulation in the outer core, we can drive it by this crystallization. This is current, ongoing research by people smarter than me, so the details may vary. But the heart of the matter is that downward-falling "snow" will displace hot fluid on the surface of the inner core, which is forced to rise. So, if what I just wrote made sense, then it is now the process that's growing our inner core that is powering the currents which make Earth's magnetic field--which protects us from solar radiation.

Which has all been a very long way around to say that the temperature/physical state (solid vs. liquid) of the core does indeed affect conditions on the surface.