Solution:

Since $720^2 = ab,$ the common ratio $r = \tfrac{720}{a} = \tfrac{b}{720}$ is rational. Write $r = \tfrac{p}{q}$ in lowest terms with $p \gt q.$ Then $a = \tfrac{720q}{p}$ and $b = \tfrac{720p}{q}$ are integers, which forces $p \mid 720$ and $q \mid 720.$ To make $b$ smallest, we want the smallest ratio $\tfrac{p}{q} \gt 1$ with both $p, q \mid 720,$ which is $\tfrac{16}{15}.$ That gives $b = 720 \cdot \tfrac{16}{15} = 768$ (and $a = 675$). The digit sum is $7 + 6 + 8 = 21.$ Thus, E is the correct answer.