Please refer this code
HTML
<asp:TextBox ID="TextBox1" runat="server" TextMode="MultiLine" Rows="10"></asp:TextBox> 
<asp:ListBox ID="ListBox1" runat="server" Height="165"></asp:ListBox>
<br />
<asp:Button Text="Add Multiline TextBox values to ListBox" OnClick="AddNewLineChacters"
    runat="server" />
C#
protected void AddNewLineChacters(object sender, EventArgs e)
{
    string[] strCharacters = this.TextBox1.Text.Trim().Split('\n');
    string temp = string.Empty;
    foreach (string s in strCharacters)
    {
        temp = s.Replace("\r", "");
        this.ListBox1.Items.Add(new ListItem(temp, temp));
    }
}
Screenshot
