Is it possible to overdose on silica?

It is about impossible to overdose on silica since it is so poorly absorbed.  Excess silica simply passes through the body.  But taking excess is also a waste for the same reason.

Does silica compete with other minerals like how copper competes with zinc?

Silica itself is not going to compete with other minerals.  In fact, it aids in the absorption of calcium.

The calcium in diatomaceous earth (DE) though will block the absorption of most if not all iron found in some DE products.