<p><a href="http://hessian.caucho.com" rel="nofollow noreferrer">Hessian</a> is a custom binary serialization protocol, (which is open-source - I think), that forms the basis for a binary cross platform remoting framework. I'd like to know if anyone here has used it, and if so, what sort of performance can we expect from a solution that bridges a Java app on one side with a C# app on the other. (Let us consider that we are serializing simple classes, and may be arrays, lists, dictionaries of simple classes.)</p>

This is the sort of problem that web services were designed to solve.  Although no longer simple, the SOAP format allows you to serialize objects to an XML representation on a Java/C# application, transmit them across the wire and deserialize them in the corresponding Java/C# application (Java/C# may be replaced with virtually any language that can translate an XML document).  Although "serialize" is used here, it is also common for this process to be referred as "marshalling".
However, moving away from SOAP for web services is currently being considered by many.  Find out more about web services from Wikipedia:
http://en.wikipedia.org/wiki/Web_services