If I have a RMD equal to 1/4 of my annual IRA withdrawals & I pay fed tax on this monthly, must I pay more tax on this again ?

It depends.

The tax that you are paying monthly is tax withheld.

Tax withheld is just an estimated advance payment of your taxes. The final tax amount can only be determined when you fill out your tax return. The taxes withheld serve as a credit against your tax liability. If too much tax was withheld, you'll receive a refund; otherwise, there'll be a tax due.

You are having tax withheld from the distributions from the retirement account, just like when an employer withheld tax from your paychecks.  When you enter it all into the software, the calculations are done to reconcile the amount of income you received and the amount of tax that was withheld.  You might owe more or you might get a refund.

Your income tax is only calculated on your tax return. Withholding is an estimate.  You must enter the 1099-R as-is.  Your tax is calculated from your overall income and deductions, and you get credit for the withholding.  If you had extra tax taken out, that comes back as a refund.  You are never "taxed twice" on retirement income.  

The reason Turbotax asks for the amount of your RMD is that there are certain things you can't do with your RMD, like roll it over into another IRA.  Asking the question helps the program figure out those other issues, but does not result in new tax.