How does one test a function that has no return value?

I would posit that if it's truly not testable, then it doesn't actually do anything at all, and shouldn't exist ;) Something along the lines of this might work...

Assert.IsNotNull( Logger.File );
Logger.CloseCurrentTextLogFile();
Assert.IsNull( Logger.File );

Or check the status of Logger.FileOpenStatus or check that Logger.OpenFile(fname) throws an exception before closing, but not after. There's got to be something in Logger whose behaviour depends on whatever action CloseCurrentTextLogFile() performs.

Static state methods naturally make themselves fairly untestable, so my suggestion is based around refactoring your code away from static methods.

I would turn Logger into an instance class that takes an IO object in the constructor. That will allow you to stub the IO object and you can Assert that your IO object's Close method was called.

This is only if you want to make your code 100% testable. Otherwise, I would agree with Mo that if it is not testable, then do not write a forced test...those tend to be very brittle. In the end, you need to be pragmatic about your code. Often a logger is useful to keep static, however as I already mentioned, these tend to be very untestable....so just be pragmatic about your work and don't write tests in the mere effort to get 100% code coverage...that 100% will come with a price...

UPDATE

Here is why this is not truly testable from a dogmatic POV of unit testing. You are not testing a unit of work, but instead you are testing the Logger AND the Logger's dependencies (the IO object in this case). It also makes your tests slower and requiring environmental setup and state (you must have first opened an actual file to close it, right?). These are all bad for unit testing, but ok for integration testing...so it depends on what kind of tests you are writing, also.

You can check the state of Logger or you could call some other method on logger that will not produce an error because you called this method, which should succeeded if you had not called the method.