The answer is yes, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:
Theorem. Successive barycentric subdivisions of a non-flat triangle contain triangles which, to within a similarity, approximate arbitrarily closely any given triangle.
Here is a nice divulgative article on the dynamics of iterated barycentric subdivisions. It also has a list of further readings on the topic. (The joy of barycentric subdivision, by Bill Casselman)-
[1] I. Barany, A. Beardon and T. Carne, "Barycentric subdivision of triangles and semigroups of Möbius maps", 1996. MR1401715.
