There's no cut off age that you stop paying income tax on social security benefits if you're making other income.
However A person only has to pay income tax on social security benefits if the retired person is making other income through self employment, dividends, stocks, website income etc.
In some places you're taxed on social security income and the cut off age for that is 66 so long as you're not making other income from other sources along with your social security benefits.
If you're not making other income and are just living off of social security benefits then that income is not taxable after you turn 66 years old.