Ygtjki

I have used an online service to do my taxes and they have asked me to mail the signed returns along with my W2s and other documents to IRS at Austin and the treasury dept of New Jersey for the state returns. I have moved to Italy and I was wondering if I can use UPS to send my tax returns.

Edit:

I permanently moved out of USA. I am filing as a non resident alien on the income I earned while I was in USA.

Edit: Maybe I should call IRS and ask them.

From the IRS website -

Austin - Internal Revenue Submission Processing Center
3651 S IH35
Austin, TX 78741

You are not the first person needing to ship to IRS via Fed-Ex or other private carrier.

For New Jersey, the shipping address is:

State of New Jersey
Division of Taxation
Revenue Processing Center
200 Woolverton Street
Building 20 
Trenton, NJ 08611

This is not a direct answer to your question, but I am suggesting an alternative. FreeFillableForms.com is endorsed by the IRS (https://www.irs.gov/e-file-providers/before-starting-free-file-fillable-forms) and allows you to e-file, which is the arguably fastest, safest way to get your return to the IRS (I am surprised your online service did not give you that option). They have most forms available, but in case you need exotic ones, double check before your start. Be aware that the IRS does some sanity checks on e-filed returns before accepting them, so you might get an e-mail a day after filing that your return was rejected because you forgot to check some box or left something blank. In that case simply correct the issue and e-file again. (I mention this so you allow a couple of days of buffer before the due date.)

Many states have similar online filing services, although some have restrictions with regard to the forms that are available (e.g. resident vs nonresident).

In your case the disadvantage is that you have to type in all the data from the forms you already have, so you are trading labor for cost savings and speed.

If you are worried about the speed of the postal service and your are expecting a refund (if you owe, the IRS might charge late fees and interest), you could also file an extension electronically (with minimal typing), so it won't matter if the return arrives a couple of days late.
EDIT: As @DilipSarwate mentioned, you have an automatic extension until June 15 anyway, so the above only applies if you needed more time than that.

I successfully filed Federal, NY and NJ taxes from overseas this year. I just went to my regular post office and posted them like I would any other normal international mail. It took around 2 months from posting to receiving my refund.

Federal:

Department of the Treasury
Internal Revenue Service
Austin, TX 73301-0215 USA

New Jersey:

State of New Jersey
Division of Taxation
Revenue Processing Center
P.O. Box 555
Trenton, NJ 08647-0555

New York:

State Processing Center
P.O. Box 61000
Albany, NY 12261-0001

Although you are not filing a NYS refund, I want to add this in case someone else who is filing NYS from overseas finds this answer. They may note that the first page of your NYS refund has this big scary message:

This message is pretty much rubbish. I have never once filed my NYS tax return electronically due to my wife not having a social security number, which immediately excludes you from e-filing. Also they do not allow e-filing when you have an overseas address. You can safely ignore this message.