It is commonly believed that tidal interactions between massive bodies can help keep their interiors molten, so regardless of whether the Moon directly influences the rotation of the Earth's core, it's certainly important to our magnetic field. Without the Moon, it is possible that Earth's core and mantle would've cooled too much to be conductive.

The most compelling evidence of this is the Galilean moons of Jupiter. Io, under the most strain from Jupiter and its larger cousins (especially Europa), is fantastically active volcanically. Europa is also obviously geologically active, although not as much. As you go further out, there is less tidal strain and the moons appear to be less active (with less resurfacing). An extreme example of tidal stress is Neptune's largest moon, Triton. It orbits retrograde, which means it goes around Neptune against Neptune's rotation. It is also active, with dramatic nitrogen geysers. Enceladus is a bit of a puzzle; there is dispute as to whether or not there is enough tidal force to keep it active, and yet it clearly has a very young surface. So there is very active debate over what keeps it warm.