C# how to have a timer close a winform

MessageBox won't let you do that, because it's a Modal form - your UI thread is effectively suspended until the you closes the MessageBox so your timer will not function.

Instead of using a MessageBox, you need to create your own form which contains it's own Timer, and which closes itself when that expires. You then display that from your first form using myNewForm.ShowDialog.

For some reason Ralfs solution did not work when I tested it, here is my modified version.
I would not recommend using the forms Timer btw. it can lead to problems with e.g. threading, better use a system Timer.

using System;
using System.Windows.Forms;

namespace WinFormsTimer1
{
    public partial class Form1 : Form
    {
        private int[] myTimer = new[] { 10, 1, 3 };

        public Form1()
        {
            InitializeComponent();

            // Timer to Close App
            System.Windows.Forms.Timer MyTimer = new System.Windows.Forms.Timer();
            MyTimer.Interval = (5000); // 5 seconds
            MyTimer.Tick += new EventHandler(Timer1_Tick);
            MyTimer.Start();
        }

        private void Form1_Load(object sender, EventArgs e)
        {
            // Initializing myTimer here does not work ???
        }

        private void Timer1_Tick(Object myObject, EventArgs myEventArgs)
        {
            this.label1.Text = myTimer[0].ToString() + " " + myTimer[1].ToString() + " " + myTimer[2].ToString();

            if (myTimer[0] > 0)
            {
                myTimer[0] -= 1;
            }
            else if (myTimer[1] > 0)
            {
                myTimer[1] -= 1;
                DialogResult Result = MessageBox.Show("The form will now be closed.", "Time Elapsed", MessageBoxButtons.OKCancel);

                if (Result == DialogResult.Cancel)
                {
                    myTimer[0] = 10;
                    myTimer[1] = 1;
                    myTimer[2] = 3;
                }
                else
                {
                    this.Close();
                }
            }
            else if (myTimer[2] > 0)
            {
                myTimer[2] -= 1;
            }
            else if (myTimer[2] == 0)
            {
                SendKeys.Send("\r");
            }
        }
    }
}

And in case you want to try the solution without a MessageBox:

using System;
using System.Drawing;
using System.Windows.Forms;

namespace WinFormsTimer1
{
    public partial class Form1 : Form
    {
        System.Windows.Forms.Timer MyTimer = new System.Windows.Forms.Timer();

        private Form form2 = new Form();

        bool form2Shown;

        public Form1()
        {
            InitializeComponent();

            // Timer to Close App
            MyTimer.Interval = (5000); // 5 seconds
            MyTimer.Tick += new EventHandler(Timer1_Tick2);
            MyTimer.Start();
            this.label1.Text = MyTimer.Interval / 1000 + " seconds remaining";
        }

        /// <summary>
        /// Closes the main form and form2 on the second Tick if needed.
        /// </summary>
        private void Timer1_Tick2(Object myObject, EventArgs myEventArgs)
        {
            if (form2Shown)
            {
                this.Close();
            }
            else
            {
                ShowForm2("Time Elapsed", "The form will now be closed.");
                form2Shown = true;
            }
        }

        /// <summary>
        /// MessageBox replacement.
        /// </summary>
        private void ShowForm2(string caption, string message)
        {
            form2.StartPosition = FormStartPosition.CenterScreen;
            form2.ClientSize = new Size(300, 200);
            form2.TopMost = true;
            form2.Text = caption;

            Label label1 = new Label();
            label1.Location = new Point(50, 50);
            label1.AutoSize = true;
            label1.Text = message;

            Button buttonOk = new Button();
            buttonOk.Location = new Point(140, 150);
            buttonOk.Anchor = AnchorStyles.Bottom;
            buttonOk.Text = "Ok";
            buttonOk.Click += new EventHandler(CloseForm);

            form2.Controls.Add(label1);
            form2.Controls.Add(buttonOk);
            form2.Show();
        }

        private void CloseForm(Object myObject, EventArgs myEventArgs)
        {
            this.Close();
        }
    }
}