What methods exist for returning all points within an arbitrary closed shape? I have been reading about range search algorithms, but most that I read apply only to a simple range (such as a standard rectangle in 2d).

In a similar fashion, what options exist for range search algorithms across arbitrary closed shapes?  For example, if I had a binary image of a person, and random points across the image, how do I determine which points lie on the portion of the person?
There are no options.

For any set of $n$ points, if you want to answer range queries for arbitrary ranges, all $2^n$ subsets are possible answers.  (In fact, for any set of $n$ points in convex position, every subset is the intersection with some convex range.)  It follows immediately that any query algorithm (describable as a decision tree) requires $\Omega(n)$ time, no matter how much time and space you spend preprocessing the points.

Equivalently: The range space (points in the plane, convex regions in the plane) has unbounded VC-dimension, so there's no hope for an efficient range searching algorithm.