Because of the graduated tax tables, the rich pay more than their fair share. A fair share is a flat percentage. And the fairest way to collect taxes it a flat consumption tax. A tax on income also gives the gov the intrusive ability to modify behavior through tax credits and deductions.
Distribution of Major Federal Taxes 2000
51% of Americans Pay No Federal Income Taxes - Derek Thompson - Business - The Atlantic
But to respond to the original question, "Should a NON-taxpayer Be Allowed to Vote?", if we had a federal consumption tax instead of a federal income tax, the question would be moot; everyone would be a tax payer. However, if we don't repeal the 16th Amendment, and more than half of the population pays no income tax and can still vote, the balance of voters has tipped to those who would redistribute the wealth of others. A democracy can only exist until the voters discover that they can vote themselves largesse from the public treasury.
Disallowing the right to vote to those who pay no taxes might stem the tide toward socialism, something we must do because, as Margaret Thatcher once said, you eventually run out of other people's money.