A positively charged metal sphere, A, is held close to but not touching and identical uncharged sphere, sphere B. Sphere A is now removed. After sphere A has been removed, sphere B is touched to an initially uncharged sphere, sphere C. What is the sign of the charge, if any, on sphere C after it has been touched by sphere B

The sphere C carries no net charge.

Explanation:

When brougth close to the charged sphere A, as charges can move freely in a conductor, a charge equal and opposite to the one on the sphere A, appears on the sphere B surface facing to the sphere A. As sphere B must remain neutral (due to the principle of conservation of charge) an equal charge, but of opposite sign, goes to the surface also, on the opposite part of the sphere. If sphere A is removed, a charge movement happens in the sphere B, in such a way, that no net charge remains on the surface. If in such state, if the sphere B (assumed again uncharged completely, without any local charges on the surface), is touched by an initially uncharged sphere C, due to the conservation of charge principle, no net charge can be built on sphere C.