Solution:

Three vertices determine a plane that cuts through the interior unless all three lie on a single face.

Each of the $6$ faces gives $\binom{4}{3} = 4$ triples, so $6 \cdot 4 = 24$ triples lie on a face out of $\binom{8}{3} = 56$ total.

The probability of hitting the interior is $$1 - \dfrac{24}{56} = \dfrac{4}{7}.$$

Thus, the correct answer is C.