How can I read a text file without locking it?

You need to make sure that both the service and the reader open the log file non-exclusively. Try this:

For the service - the writer in your example - use a FileStream instance created as follows:

var outStream = new FileStream(logfileName, FileMode.Open, 
                               FileAccess.Write, FileShare.ReadWrite);

For the reader use the same but change the file access:

var inStream = new FileStream(logfileName, FileMode.Open, 
                              FileAccess.Read, FileShare.ReadWrite);

Also, since FileStream implements IDisposable make sure that in both cases you consider using a using statement, for example for the writer:

using(var outStream = ...)
{
   // using outStream here
   ...
}

Good luck!

Explicit set up the sharing mode while reading the text file.

using (FileStream fs = new FileStream(logFilePath, 
                                      FileMode.Open, 
                                      FileAccess.Read,    
                                      FileShare.ReadWrite))
{
    using (StreamReader sr = new StreamReader(fs))
    {
        while (sr.Peek() >= 0) // reading the old data
        {
           AddLineToGrid(sr.ReadLine());
           index++;
        }
    }
}