On page 221 of The Precipice, Toby Ord says "scientists expect the brightening of the sun to also slow Earth's plate tectonics, dampening volcanic activity". This seemed very weird to me; how could sunlight possibly have any effect on mantle movement?

The book didn't provide any direct citations for this statement, but a related footnote mentions Kellogg (2013), Wolf & Toon (2015), Leconte et al. (2013), and Rushby et al. (2018).

I couldn't find Kellogg. Wolf & Toon doesn't mention anything about this. There were multiple possible papers for Leconte et al., none of which seemed relevant. The most promising one was Rushby et al., which mentions:

The importance of mantle hydration as a mechanism for facilitating and maintaining plate tectonics over geological time remains unclear (Korenaga, 2010; Korenaga, 2012). It is possible that a drier mantle and lower associated viscosity will make the initiation of plate tectonics easier, but that maintaining a continental crust may require interaction between the hydrosphere and solid Earth over geological time.

Korenaga (2010) does mention this in more detail, saying:

a mechanism based on the presence of surface water has the advantage of being able to readily explain the absence of plate tectonics on other terrestrial planets in the solar system.

Korenaga (2012) doesn't seem to be available online, but the abstract mentions the same thing. And going some Googling turns up papers like this one, which says:

In the ductile creep regime, the viscosity of wet rocks is weaker than the viscosity of dry rocks by several orders of magnitude. In the brittle regime, the most substantial effect is probably serpentinization which can reduce the friction coefficient by a factor of 2 or more. The difference between the strength of a wet lithosphere and that of a dry lithosphere seems to be big enough to control the very existence of plate tectonics. Because of dehydration due to partial melting the oceanic lithosphere is expected to be essentially dry above some critical depth, around 60-80 km. This would make the lithosphere strong enough to prevent plate motion. Percolation of water from the surface can be the main mechanism supplying water to the upper parts of the lithosphere. This implies that liquid water can be crucial for maintaining plate tectonics.

And many others that confirm this picture, like The fate of water within Earth and super-Earths and implications for plate tectonics and this weird study/article hybrid from ScienceDaily.

So surprisingly enough this does seem to be a real thing.