Is $[0,1]^\omega$ homeomorphic to $D^\omega$?
The answer is yes, they are homeomorphic.
Edit: Here's a better way to put the answer below:
Theorem (Klee, 1955). Let $K$ be a compact, convex and metrizable set in a locally convex space $E$. If $K$ is not contained in a finite-dimensional subspace of $E$ then $K$ is homeomorphic to the Hilbert cube $[0,1]^\omega$.
This applies directly to the compact convex set $K = D^\omega$ in the metrizable space $E = \mathbb{R}^\omega$. This result is not explicitly stated this way in Klee's work [1], but it follows immediately from Theorem (1.2) of that paper. The proof idea is outlined a bit further down.
Original Answer:
Klee proved in 1955, based on a theorem due to Keller, the following remarkable result (Theorem (1.2) of reference [1] below):
Let $X$ be a separable normed space and let $E$ be either $X$ with the norm topology, $X$ with the weak topology or $X^\ast$ with the weak$^\ast$ topology. If $K$ is an infinite-dimensional compact convex subset of $E$ then $K$ is homeomorphic to the Hilbert cube $[0,1]^\omega$.
The idea is that in each case one can find a countable family of continuous linear functionals $(f_n)$ separating the points of $K$ and, normalizing them appropriately, the map $x \mapsto (f_1(x),f_2(x),\ldots)$ gives a linear homeomorphism from $K$ onto an infinite-dimensional norm-compact convex subset of $\ell^2$. Keller had previously shown that all infinite-dimensional compact convex sets in $\ell^2$ are homeomorphic to $[0,1]^\omega$.
Since $D^\omega$ is a compact convex subset of $\ell^1 = (c_0)^\ast$, Klee's theorem applies. To finish up, note that the topology on $D^\omega$ viewed as a subset of $(c_0)^\ast$ and the topology induced from $\mathbb{R}^\omega$ coincide by the standard proof of Alaoglu's theorem.
Here are the relevant papers:
V. L. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30–45. MR0069388
Ott-Heinrich Keller, Die Homoiomorphie der kompakten konvexen Mengen im Hilbertschen Raum, Math. Ann. vol. 105 (1931) pp. 748–758. MR1512740