The stimulus payments are not taxable income. You will not get a 1099 form for them. At the beginning of the Federal Review TurboTax will ask whether you received a stimulus payment, and if so, how much. This is to make sure that you get the full amount that you are entitled to.

The IRS will mail you a notice #1444 (first check)  and/or   #1444-A (second check)  for the stimulus amounts you did receive in advance and are used to complete the 2020 return so you have the exact amount needed when you are asked.  

The stimulus checks are  an advance on a credit you can receive on your 2020 tax return.  If something went wrong or you did not get the correct amount of both  stimulus payments by 1/27/2021 then  you can get it when you file your 2020 return in early 2021—if you are eligible.  It will end up on line 30 of your 2020 Form 1040.

https://www.irs.gov/newsroom/economic-impact-payment-information-center-topic-j-reconciling-on-your-...

https://www.irs.gov/coronavirus/coronavirus-tax-relief-and-economic-impact-payments-for-individuals-...

https://ttlc.intuit.com/community/tax-topics/help/how-will-the-stimulus-package-impact-me/00/1393859

At this point, we do not know yet how long  the new stimulus bill will  delay the filing season due to changes to the 2020 Form 1040, software programming, software updates, etc., etc.  Do NOT be in a hurry to file your 2020 tax return until more is known. Give some extra time to the IRS and the software programmers so that the software can handle the new 2nd stimulus, so that you can get it all by filing ONE time. 

If I do not enter the Stimulus amount, I get a refund. Once I go back and enter the Stimulus then my refund drops and now I owe taxes. I didn't think you had to pay taxes on the Stimulus amount. Any ideas?

Before you enter the amount of the stimulus payments at the screen Did you get a stimulus payment?, you have been given the full Rebate Recovery Credit (the equivalent of the two stimulus payments).   This can be found on line 30 of the 2020 Federal tax return.  At this point, this is an 'incomplete' Rebate Recovery Credit.

You can see this on the screen Great news! You qualify for the stimulus credit.  The amount is displayed in the center of the page.

After you enter stimulus payment 1 and stimulus payment 2, the credit is offset by the stimulus payments and the 'incomplete' Rebate Recovery Credit is reduced or eliminated.

See this TurboTax Help.

This TurboTax Help states that the Economic Impact Payments (both the first and second) are not considered income, and therefore are not taxable.

However, the state of Oregon is taxing stimulus payments as income.