The only tax Amish and Mennonites are exempt from is the Social Security tax, and only if they are self-employed. For the most part, they pay the same taxes that other Americans do.
Amish people pay all sales taxes. They also pay income taxes to both the federal government and the state, along with property taxes and other types of taxes. They have a religious objection to Social Security because they believe that the Bible commands them to take care of their elderly as they see fit. For the most part, the Amish keep elderly people in their own homes to provide care.