I live in NYC. If I sell my house which I've owned for 50 years on March 15th, 2014, when is my capital gains tax payment due?

The calender year the gains were realized... meaning when the sale went to close.

If the gain is substantial, and assuming that at least some portion of it does not qualify for the homeowner's capital gain exemption (i.e., the house was not your principal residence for at least 2 of the previous 5 years, and/or the gain exceeds $250K, or $500K if married filing jointly), you should consider paying the capital gains tax due with a quarterly estimated tax filing. Reason is that if you wait until April 2015 to pay the tax you may get hit with interest penalties for having underpaid taxes during the year. The first quarter estimated tax payment is due April 15.

Ebaines has it right. A house that you have owned for 50 years in NYC has probably appreciated by a factor of 100, so the capital gains WILL be substantial. A quaterly estimate tax payment most definitely WILL be needed.