Does anyone know for which values of $D$ the equation $X^3=DY^3+A^3$ has solutions? All numbers non-zero naturals.
The answer is that nobody knows! An answer to your question is equivalent to asking what numbers $D$ can be expressed as the sum of two rational cubes $u^3+v^3=D$, where $u=X/Y$ and $v=-A/Y$, but the classification of such numbers $D$ is not known. (If one believes the Birch and Swinnerton-Dyer conjecture, then there is an analytic method to test that will check whether $D$ is a sum of rational cubes, though.)
This article by J. H. Silverman is a great introduction to the subject. I can't recommend it highly enough.