.NET Fiddle

CSharp · Console

/Program.cs

using System;
using System.Collections.Generic;
using System.Text;
using System.Text.RegularExpressions;

public class Program
{
    public static void Main()
    {
        string pattern = @"([""'])(?:(?=(\\?))\2.)*?\1";
        List<string> lines = new List<string>(); 
        lines.Add("4121,6383,0,,,TRUE");
        lines.Add("4122,6384,0,\"20,000 BONUS POINTS\",,TRUE");
        lines.Add("4123,6385,,,,");
        lines.Add("4124,6386,0,,,TRUE");
        lines.Add("4125,6387,0,,,TRUE");
        lines.Add("4126,6388,0,,,TRUE");
        lines.Add("4127,6389,0,,,TRUE");
        lines.Add("4128,6390,0,,,TRUE");

        StringBuilder sb = new StringBuilder();
        foreach (var line in lines)
        {
            sb.Append(Regex.Replace(line, pattern, m => m.Value.Replace(",", ""))+"\n");
        }           
        Console.WriteLine(sb.ToString());
    }
}