Does faster exact algorithm for counting independent sets in comparability graphs than general graph exisits?
Bipartite graphs are comparability graphs, and it's known (since Provan and Ball) that counting the number of independent sets in bipartite graphs is #P-complete.

J. S. Provan and M. O. Ball. The complexity of Counting Cuts and of Computing the Probability that a Graph is Connected.  SIAM J. Comput. 12 (1983) 777-788.

If I haven't mistaken, this should answer your question.