Ask Ubuntu Data Server does what you suggest in a very small way. It uses the information in the data dumps to respond to the
/similar route. Its behavior is designed to match the actual API route as much as possible.
That being said, it serves mostly as a demonstration that such a thing is possible. It only implements one route and only a fraction of the parameters for it. It also does not use the data dumps as they are, but rather precomputes an index file that allows for faster lookups / searches. (Not only that, but it actually uses the real API to keep the information up to date.)
So if you're looking for an application that uses the data dumps to provide a good portion of the API, I'm afraid no such thing exists right now (at least nothing listed here on Stack Apps). But if you're looking for an example to get started with, Ask Ubuntu Data Server is probably the closest thing.
On a completely different note, perhaps something like this could be easily constructed by using Sky Sander's SQL Data Dump Import tool to stick the contents of the data dump into an SQL database. Then all you need to do is come up with SQL queries that reproduce the behavior of the real API and write a front-end that returns the result as JSON data.