If I entered the high deductible health insurance premium amount from box 14 while doing my federal taxes into TurboTax, why did the George State Tax forms not flag that I entered it again on my State Tax forms? Isn't it double dipping if it was paid with pretax dollars from my pay check?
Box 14 on the W-2 is an "Other " box, where the entries are not necessarily codified, but is used by the employer to indicate something to the employee. See page 21 for the instructions for line 14 at https://www.irs.gov/pub/irs-pdf/iw2w3.pdf
Normally, employer payments (and your part by means of payroll deduction) towards health insurance premiums (your HDHP) are shown in box 12 of the W-2 with a code of DD. In addition , this amount is removed from Wages because it is pre-tax.
Because Box 14 entries are not codified (the employer could make up whatever he/she wants here), TurboTax can't use them to carry amounts to other parts of the return.
TurboTax has no way of knowing if the box 14 is pre-tax - it can know this only if it was put in box 12, where the codes are well-defined and listed in the IRS instructions.
It was good that you caught this. You might also ask your employer why this amount is not shown in box 12 with code DD and to confirm that the amount is indeed pre-tax.