I got a bit puzzled when removing the vertex groups from the teeth to leave them only influenced by the Head Bone. This left the teeth stretched right through to the back of the head.
I selected the armature, then the teeth, and went into weight paint mode, and after removing the other bones from the vertex groups, it seemed that some of the vertices on the teeth were now not influenced by anything at all. This left these vertices in the same position throughout the animation. I manually weight painted these vertices and it seems to have fixed the issue.
I’m aware that this might just be an issue with my topology, but thought I’d post this here in case anyone else comes across this issue.