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++;
        }
    }
}

new StreamReader(File.Open(logFilePath, 
                           FileMode.Open, 
                           FileAccess.Read, 
                           FileShare.ReadWrite))

-> this doesn't lock the file.

The problem is when you are writing to the log you are exclusively locking the file down so your StreamReader won't be allowed to open it at all.

You need to try open the file in readonly mode.

using (FileStream fs = new FileStream("myLogFile.txt", FileMode.Open, FileAccess.Read, FileShare.ReadWrite))
{
    using (StreamReader sr = new StreamReader(fs))
    {
        while (!fs.EndOfStream)
        {
            string line = fs.ReadLine();
            // Your code here
        }
    }
}