In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another direction there is a bound, and this is discussed in the nice paper below (on the first page):
Le Donne, Enrico; Rajala, Tapio, Assouad dimension, Nagata dimension, and uniformly close metric tangents, Indiana Univ. Math. J. 64, No. 1, 21-54 (2015). ZBL1321.54059.