To prove that in $\triangle ABC$ with $AB \neq AC$, it follows that $\angle B \neq \angle C$, we approach this by proof by contradiction. The essence of proof by contradiction involves assuming the negation of what we want to prove and showing that this assumption leads to a contradiction.

Given that we want to prove $\angle B \neq \angle C$, the direct negation of this statement is $\angle B = \angle C$. This means, to use proof by contradiction effectively, we assume that $\angle B = \angle C$ and show that this assumption contradicts the given conditions or leads to an impossible scenario within the context of $\triangle ABC$ where $AB \neq AC$.

Therefore, the correct assumption to make, based on the method of proof by contradiction, is:

- A: $\angle A = \angle B$ (Incorrect, does not directly negate the statement we want to prove)
- B: $AB = BC$ (Incorrect, does not directly relate to the angles at vertices B and C)
- C: $\angle B = \angle C$ (Correct, directly negates the statement $\angle B \neq \angle C$)
- D: $\angle A = \angle C$ (Incorrect, does not directly negate the statement we want to prove)

Hence, the correct assumption to make for the proof by contradiction is $\angle B = \angle C$.

So, the answer is encapsulated as $\boxed{C}$.