Answer: No, you do not have to pay any taxes on any stimulus check you received in 2020.
The stimulus checks, officially is called the economic impact payments, were a tax credit for your 2020 taxes. The stimulus check will not increase your income nor your taxes.
The first round of the Cares Act gave up to $1,200 to each adult taxpayers who earned under $75,000 per year. In the second round, each taxpayer received $600. The purposed third round of the Cares Act will be an additional $1,400 per adult taxpayer.