Based on my conversations with conservatives, I think most of them also believe that the affluent have a moral obligation to help the less fortunate. They simply dispute that the government has a right to force them to help via taxes.
One example would be Christian conservatives, who often believe that their faith requires them to help the less fortunate. But there are also many conservatives who believe that no such obligation exists, that it's each man to himself and survival of the fittest. Strangely enough, I know some Christians who are also social darwinists, which makes you wonder what their Christianity actually means.