How to suppress a dialog box displayed by code that I can’t change?

A message box pumps a message loop. That’s something you can take advantage of, it allows you to inject code with Control.BeginInvoke() that runs as soon as the message box appears. You can then use that code to find the dialog window and close it. Add a new class to your project and paste this code:

using System;
using System.Text;
using System.Windows.Forms;
using System.Runtime.InteropServices;

public class DialogCloser : IDisposable {
    public DialogCloser() {
        if (Application.OpenForms.Count == 0) throw new InvalidOperationException();
        Application.OpenForms[0].BeginInvoke(new Action(() => {
            // Enumerate windows to find dialogs
            if (cancelled) return;
            EnumThreadWndProc callback = new EnumThreadWndProc(checkWindow);
            EnumThreadWindows(GetCurrentThreadId(), callback, IntPtr.Zero);
            GC.KeepAlive(callback);
        }));
    }

    public void Dispose() {
        cancelled = true;
    }

    private static bool checkWindow(IntPtr hWnd, IntPtr lp) {
        // Checks if <hWnd> is a Windows dialog
        StringBuilder sb = new StringBuilder(260);
        GetClassName(hWnd, sb, sb.Capacity);
        if (sb.ToString() == "#32770") {
            // Close it by sending WM_CLOSE to the window
            SendMessage(hWnd, 0x0010, IntPtr.Zero, IntPtr.Zero);
        }
        return true;
    }

    private bool cancelled;

    // P/Invoke declarations
    private delegate bool EnumThreadWndProc(IntPtr hWnd, IntPtr lp);
    [DllImport("user32.dll")]
    private static extern bool EnumThreadWindows(int tid, EnumThreadWndProc callback, IntPtr lp);
    [DllImport("kernel32.dll")]
    private static extern int GetCurrentThreadId();
    [DllImport("user32.dll")]
    private static extern int GetClassName(IntPtr hWnd, StringBuilder buffer, int buflen);
    [DllImport("user32.dll")]
    private static extern IntPtr SendMessage(IntPtr hWnd, int msg, IntPtr wp, IntPtr lp);
}

Sample usage:

private void button1_Click(object sender, EventArgs e) {
    using (new DialogCloser()) {
        // Replace this with the call to the COM server method:
        MessageBox.Show("you never see this");
    }
}