When i run a simple test on connection to DB check i receive an error in NUnit:
[Test]
public void TestConn()
{
string connectionString = Configuration
Your unit tests should still work as long as you have the same configuration for your test project as for your main project.
I'd suggest using a pre-build event in your test project to copy your application's configuration file over to the test project. This saves having to maintain two sets of configuration.
copy $(SolutionDir)path-to-main-project\Web.config $(ProjectDir)App.config