I can't see it being a blocking problem.

Looking again at the code, I think what it might be is that you're not closing the response and hence the connection and after a while you're running out of connection handles!

See if it's any better if you rearrange the code as follows:

private static void HTTPCheck(string url)
{
    HttpWebResponse resp = null;
 
    try
    {
        HttpWebRequest req = (HttpWebRequest)HttpWebRequest.Create(url);
        resp = (HttpWebResponse)req.GetResponse();
        switch (resp.StatusCode)
        {
            case HttpStatusCode.OK:
                //ok
                break;
            default:
                //failed
                break;
        }
    }
    catch (Exception x)
    {
        WriteToLog(null, null, null, x.Message);
        Console.WriteLine("{0} - {1}", DateTime.Now, x.Message);
        //failed
    }
    finally
    {
        if(resp != null) resp.Close();
    }
}

The default timeout interval for HttpWebRequest is 100 seconds so, if it's a slow or busy site and you're checking every minute, it's possible that the previous request hasn't yet been responded to when a new one goes in.

I'd try checking every two minutes instead or leave it at one minute but reduce the timeout period to, say, 50 seconds before you call GetResponse():

req.Timeout = 50000;