I have paid $1,000 for property tax for fiscal year July 1,2016 - June 30,2017. How much can I claim for 2016 tax return?

[ThomasM]

Claim your property tax in the year you paid it, regardless whether it's for a calendar year or a fiscal year.  Property tax paid in 2016 gets claimed on the 2016 return.  

[Mr--W]

What if I paid $500 and the remainder is due on January 31,2017. Do I still claim the full amount?

[ThomasM]

If you paid $500 in 2016 and the other $500 in January 2017, then you'll claim only the $500 payment from 2016.