Let $(M, \partial M)$ be an orientable $n$-dimensional topological manifold with boundary. Suppose that $(N, \partial N)$ is an $n$-dimensional topological manifold with boundary and $N \subset M$.
If $N- \partial N$ is an open subset of $M$ then I believe that $N$ is also orientable since the orientation of $M$ (which can be viewed compatible choice of local orientations at each point $x \in M - \partial M$) should induce an orientation of $N$.
If $N - \partial N$ is not an open subset of $M$ then can I say if $N$ is orientable or not?
Intuitively, yes, because orientability is a local property in the first place. So I'm assuming the tangent bundle of N is a subbundle of the tangent bundle of M. You can have nonorientable subspaces of an orientable manifold because orientability is defined relative to the dimension of the space.