How to explicitly discard an out argument?

You have to declare a variable which you will then ignore. This is most commonly the case with the TryParse (or TryWhatever) pattern, when it is used to test the validity of user input (e.g. can it be parsed as a number?) without caring about the actual parsed value.

You used the word "dispose" in the question, which I suspect was just unfortunate - but if the out parameter is of a type which implements IDisposable, you should certainly call Dispose unless the method documentation explicitly states that receiving the value doesn't confer ownership. I can't remember ever seeing a method with a disposable out parameter though, so I'm hoping this was just an unlucky choice of words.

Unfortunately you are required to pass something because the method is required to set it. So you cannot send null because the method, being required to set it, would blow up.

One approach to hide the ugliness would be to wrap the method in another method that does the out parameter for you like so:

String Other_MakeMyCall(String inputParams)
{
    String messages;

    return MakeMyCall(inputParams, out messages);
}

Then you can call Other_MakeMyCall without having to fiddle with out parameters you don't need.

Starting with C# 7.0, it is possible to avoid predeclaring out parameters as well as ignoring them.

public void PrintCoordinates(Point p)
{
    p.GetCoordinates(out int x, out int y);
    WriteLine($"({x}, {y})");
}

public void PrintXCoordinate(Point p)
{
    p.GetCoordinates(out int x, out _); // I only care about x
    WriteLine($"{x}");
}

Source: https://blogs.msdn.microsoft.com/dotnet/2017/03/09/new-features-in-c-7-0/