$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness) of one necessarily imply the finiteness (or infiniteness) of another?
To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better have finite injective dimension (the converse is also quite easy).
The local rings $R$ which have finite inj. dim. over themselves are also known as Gorenstein rings. In fact, a theorem by Foxby says that $R$ possesses a module of both finite proj. and inj. dim. if and only if $R$ is Gorenstein.
No. For example, there are rings $A$ for which the residue field $k$ is of infinite projective dimension but of finite injective dimension. On the other hand, if $k$ is of finite projective dimension, then $A$ is of finite global dimension, so $k$ (and everything else) has finite injective dimension.
For a small non-commutative example, consider the algebra $A$ which is the quotient of the path algebra of the quiver
modulo the ideal generated by $\beta\alpha$ and $\beta^2$. Then the simple module supported on the vertex $1$ is an injective of infinite projective dimension. By duality, the opposite algebra has a projective module of infinite injective dimension.