Solution:

Recall that the sum of the first $N$ number is $\dfrac{N(N + 1)}{2}.$

We want to find $N$ such that $$\dfrac{N(N + 1)}{2} = 2016.$$ Cross-multiplying and simplifying gives us $$N^2 + N - 4032 = 0.$$ Factoring gives us $$(N - 63)(N + 64) = 0$$ We want the positive value so $N = 63.$ Adding together the digits gives us $9.$

Thus, the correct answer is D .