Unit of work in mongodb and C#
Don't worry too much about opening and closing connections. The MongoDB C# driver maintains an internal connection pool, so you won't suffer overheads of opening and closing actual connections each time you create a new MongoServer
object.
You can create a repository interface that exposes your data logic, and build a MongoDB implementation that is injected where it's needed. That way, the MongoDB specific connection code is abstratced away from your application, which only sees the IRepository.
Be careful trying to implement a unit-of-work type pattern with MongoDB. Unlike SQL Server, you can't enlist multiple queries in a transaction that can be rolled back if one fails.
For a simple example of a repository pattern that has MongoDB, SQL Server and JSON implementations, check out the NBlog storage code. It uses Autofac IoC to inject concrete repositories into an ASP.NET MVC app.