I agree; it may cause SUPER localized earthquakes, especially if there was some 'bubble' of earth that settled, but in the scheme of things a bit of drilling shouldn't affect the tectonic plates.
None the less, it's hard to tell what does and does not affect the shifting of those plates. It could be a super slow process, not at all related, or anything in between. The earth is one of those things that's hard to understand, especially because we've
a) been around for such a SUPER short period of time
and b) humans evolved at one of the most stable times on the earth EVER, in regards to extinction, global heating/cooling, and tectonic activity.