How much does the Yarkovsky effect depend on the overall size of an asteroid?
Is the effect negligible for really large asteroids?

The Yarkovsky effect is strongly dependent on size, but also on rotation rate, albedo and a host of other factors. Assuming everything stays constant except for size, the effect on a small asteroid is, as you suspect, more significant than for a large asteroid, due to the decreasing surface area to volume ratio as asteroid size increases (the volume, and as a result, the mass, increases by the cube of the radius, where the surface area increases by the square of the radius).

Rotation rate also tends to be higher in smaller asteroids, which will increase the significance of the Yarkovsky effect. If an asteroid has a high albedo, the Yarkovsky effect will be less significant than an asteroid with a low albedo due to the reduced absorption of light (which reduces the difference between daytime and nighttime temperatures on the asteroid).

As an example of this, the asteroids 6489 Golevka and 4 Vesta have approximately the same semi-major axis and rotation period, however Vesta is nearly 1,000 times larger in diameter than Golevka. This should result in a Yarkovsky effect approximately 1,000,000 times more significant for Golevka than Vesta with respect to orbital acceleration.