According to the New York new driver's brochure, a motorist with a 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. There does not appear to be a master list of states in which a NY permit is allowable for driving.
Here is the information we have on the states you requested information on.
The Ohio Bureau of Motor Vehicles (BMV) notes that if you have a learner's permit from another state it is legal to drive in OH. You may operate a vehicle in the state of Ohio with a valid learner's permit from another state, as long as there is a valid licensed driver, the age of 21 or over, in the front passenger seat.
[Let CarInsurance.com help you find affordable auto insurance now.] The Illinois Secretary of State (SOS) states that they accept any valid learner's permit, subject to restrictions mandated by the issuing state. You may also want to check and see if they have any extra time, passenger or cell phone restrictions you would have to follow.
The Indiana BMV says that an out of state learner's permit holder may operate a vehicle in Indiana as long as they follow the guidelines of the state in which that permit was issued. The passenger seat beside the permit driver must be occupied by a guardian or relative of the holder who holds a valid (full) driver's license.
Pennsylvania through a PennDOT representative stated that an out-of-state driver may only operate a motor vehicle in Pennsylvania (PA) if he or she has a full valid driver's license from another state, Canadian province, U.S. Possession, or U.S. Territory. An out-of-state learner's permit or the like is not considered valid in PA. So with an out of state permit you cannot drive in Pennsylvania.
The Missouri Department of Revenue notes that there is nothing in Missouri law that prevents a MO permit holder from driving while visiting other states, assuming that you continue to observe the restrictions placed on your driving by Missouri's Graduated Driver License Law.
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The MO DOR though does not clearly state whether they allow out of state permit holders to drive in their state. So we contacted them to get this information for you. The representative told us that if you are over the age of 16, you can drive in Missouri on your NY learner's permit by following the restrictions set forth by that state. Missouri though does not recognize an out-of-state instruction permit when the visiting permit holder is under the age of 16.
State laws do change and are amended so while this is the information we received from the states' DMVs you mentioned you may still want to contact any state's Department of Motor Vehicles you plan to drive through to check and make sure at your specific age you can drive with your NYS learner's permit and under what conditions and restrictions before you leave on any trip that takes you through these or other states. | 
