The short answer is that most likely NY will tax this money and NJ will not. Therefore, your withholding is probably OK.
The long answer is a bit longer:
At the end of 2015, you will file as a NY State and NYC resident for the period of 1/1/15 through 6/1/15. All income earned during this period will be taxed by both NYS and NYC. In addition, you will also file as a nonresident (the period of 6/2-12/31/15) and will be taxed (by NYS, not NYC) on any NY source income. Your NY source income is at the very least your wages. I don't know if you have any other income that must be considered.
You will also file a NJ return as a part year resident for the 6/2-12/31/15 period. All income earned (that is taxable in NJ) during this period must be reported and tax will be calculated on it. However, NJ will allow a credit for taxes that were paid to NY. Since generally speaking NY taxes are higher than NJ, the resulting affect is that NY taxes your wages and NJ does not.
Your employer may not be required to withhold taxes for NJ, and therefore there are no entries for it on your pay stub. Even if your employer was required to withhold, the withholding rules generally take this credit into account. Therefore, it is normal for a person in your situation to have only NY tax withheld.
I assume that you informed your employer of your move. You may want to check your pay stub to be sure that NYC tax is no longer being withheld. Since you are no longer a city resident, you are not subject to tax or withholding on your earnings.
As a point of clarification, the IRS has nothing to do with your employer withholding for any state.
If you wish to inform the IRS of your move, see here: https://www.irs.gov/Help-&-Resources...ddress-Changes