How can I better my URL checker program?
You forgot using using in your StreamReader

    using(StreamReader file = new StreamReader(inputFile)){
    string outputFile = @"D:\tmp\out.txt";

    ClearOutputFile(outputFile);

    while(!string.IsNullOrEmpty(line = file.ReadLine()))
    {
        string currentUrlAndStatus = CheckURL(line);
        if(!string.IsNullOrEmpty(currentUrlAndStatus))
        {
            WriteToFile(currentUrlStatus, outputFile);
        }
    }
  }


Use the var keyword for local variables, let the compiler figure out the type for you. Type inference helps in refactoring.

 var line = string.Empty;
 var inputFile = @"D:\tmp\links.txt";
 using(var file = new StreamReader(inputFile)){
 //...
 }
 var outputFile = @"D:\tmp\out.txt";
 // etc..