I work as a consultant on a1099 tax basis in Philadelphia, but, live in NJ. Do I pay the qtrly 2016 estimated tax to the City of Philadelphia or State of NJ?

The PA/NJ tax reciprocity applies only to employee compensation that is subject to withholding,  and it does not apply to city or local taxes.  Therefore your 1099-MISC earnings will be subject to PA state income tax, as well as any Philadelphia city tax.  Therefore you should make your estimated tax payments to PA and if necessary to Philadelphia.

Your 1099 income will also be taxable by your home state of NJ, but you'll be able to take a credit on your NJ return for taxes paid to PA.

References:

https://revenue-pa.custhelp.com/app/answers/detail/a_id/851/kw/PA%2FNJ%20reciprocity

http://www.state.nj.us/treasury/taxation/njit25.shtml

Thanks Tom.  If I am reading this correctly, I have to make Estimated Tax Payments to the State of PA & City of Philadelphia?  Correct?

Yes, assuming no taxes are being withheld on your 1099-MISC for either jurisdiction.

Can I receive credit towards NJ taxes for taxes paid to Philadelphia on 1099-MISC income?  Or do I just get credit for taxes paid to PA on 1099-MISC income?

Yes, you can take a credit on your NJ return for the Philadelphia wage tax.

That also applies to taxes paid to Philadelphia on 1099 MISC income?  I am also taking a credit on my NJ return for taxes paid to PA state on my 1099 MISC income....

Yes.  If you paid Philadelphia wage tax on your 1099-MISC income, you can take a credit for it on your NJ return.