From the information we have found you can drive in Massachusetts but not New Hampshire with your New York learner's permit.
According to the New York new driver's brochure, put out by the New York Department of Motor Vehicles, if you have either a New York learner's permit, junior permit or junior license may drive outside of New York State if it is allowable by the laws of the other state(s).
If you drive out of New York, you must obey that state's learner or junior permit and license restrictions that may apply. The NYS DMV suggests that you check with the police or motor vehicle authorities of the states which you plan on visiting in your automobile.
The Massachusetts Registry of Motor Vehicles (RMV) notes on their site that you must be at least 16 years of age to drive in Massachusetts, even if you hold a valid driver's license from another state. If you live in MA you must be at least 16 and 1/2 to obtain a Massachusetts driver's license.
Since the minimum age is higher in Massachusetts than 16 we checked with the Massachusetts DMV to make certain at just 16 years of age you can drive there with your out of state (New York) permit. The Massachusetts DMV representative we contacted stated that yes, you can drive in Massachusetts at just 16 with an out of state permit. You must however follow the rules of your state and the rules of Massachusetts regarding permits she said.
In MA anyone who is operating a motor vehicle on a Learner's Permit, regardless of his or her age, must be accompanied by a licensed driver who is at least 21 years of age. This supervising adult must have at least 1 year of driving experience and occupy the front seat beside the permit driver.
A permit holder who is under the age of 18 may not operate a motor vehicle between the hours of midnight and 5 AM unless accompanied by his parent or legal guardian who must be licensed operator with at least 1 year of driving experience and whose license is not suspended or revoked. Also the holder of the permit must be in physical possession of the permit when operating a vehicle.
The Massachusetts Learner Permits and Junior Operator's License (JOL) fact sheet, however, states that the MA time and passenger restrictions do not apply to out of state teen license holders going through a graduated driver's licensing process so we would advise you to contact the Massachusetts RMV directly to find out what restrictions they do want you to adhere to while operating a vehicle in their state with your New York learner's permit.
So it appears that yes, you can drive in Massachusetts with your New York permit but you should check with both the New York DMV and the Massachusetts RMV to find out for certain what restrictions and conditions you need to follows according to the laws of each state.
As for New Hampshire, a NH Division of Motor Vehicles representative told us that they do not accept any out of state permits. So in this state your New York learner's permit would not be accepted as a type of a license and thus you cannot drive in this state with just a New York State permit.
As you know the State of New York requires that with your permit you must be accompanied by a supervising driver at least age 21 that has a license valid for operating the vehicle you are driving so you will need to follow this condition when in Massachusetts or any other state that allows you to operate a vehicle with just a permit. NH is not such a state so you will have to wait to until you are licensed.
Remember to also check with your insurance provider to make sure they will cover you out of state with you just having a learner's permit. Get teen car insurance information here and then follow this link for young drivers car insurance needs.