The poor and middle class pay plenty of taxes ...
... Just maybe not federal income taxes. But if they get a paycheck at all they're paying Social Security/Medicare taxes, they're paying sales taxes in most states and income taxes in some states. The pay property taxes either directly if they own property or indirectly if the rent their abode.
As far as the rich paying most of the taxes (Federal income tax specifically), the top 5% own more than 50% of the wealth in this country., the bottom 50% own less than 5% of the wealth. You can't get blood from a turnip.