Использование HtmlAgilityPack
string html = @"Contrary to popular belief,.....";
var doc = new HtmlAgilityPack.HtmlDocument();
doc.LoadHtml(html);
foreach (var span in doc.DocumentNode.Descendants("span"))
{
var style = span.Attributes["style"].Value;
span.Attributes["style"].Value = String.Join(";", style.Split(';').Where(s => !s.ToLower().Trim().StartsWith("background:")));
}
var newHtml = doc.DocumentNode.InnerHtml;