Is any chordal graph an incomparability graph?
I believe the answer to your question, and to most questions like this, is to be found on http://graphclasses.org/

There's also a book that has much of this (including an appendix at the back with some of the main subset relations between graph classes): Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X.

The answer to your specific question is no. The graph shown in http://commons.wikimedia.org/wiki/File:SubdividedTriangle.png (a central triangle with three more triangle attached to its edges) is chordal, but its complement http://en.wikipedia.org/wiki/File:Forbidden_interval_subgraph.svg is not a comparability graph.