What changes do narrow-body airliners undergo to serve (relatively) long routes? I will try to pose a well-structured question here on the longest routes ever or currently served by small, narrow-body airliners - specifically, the Boeing 737 and the Airbus A319 and A320.

Set aside special versions in the likes of the Boeing 737-700BBJ, what makes a small, narrow-body airliner serve routes that are longer than those on which it is usually deployed? 

Which is to say, what is added or changed, on the design side of things, to the structure or operability of planes that are not specifically designed for long distances, to make them capable of flying longer distances? I guess the answer is ETOPS, but I am not sure of what this implies from the operational standpoint.

For my moderate understanding it is quite impressive that narrow-body birds can fly from, say, Tokyo Narita to Mumbai - a distance of approximately 6,800 km or 4,220 mi (All Nippon Airways Boeing 737-700ER, flight NH829). Another example is Sunwing Airlines flight WG481 from Vancouver to Punta Cana (a Boeing 737-800 which flew for 5,930 km or 3,690 mi).

A list of such unusual performances can be seen here.
The 737 ER (Extended Range) differs from the rest of its family by having an extra fuel tank extending its effective range.

Legally speaking (for US registered aircraft) you don't need to carry life preservers for each passenger, if you are only flying over land routes. So if a plane is being used for say a JFK $\rightarrow$ Nashville route, they would not be required to carry life preservers; but if they decided to use the plane on an over water route, they would have to outfit it as such. 

Inter-Country travel may also effect the outfitting of a plane, as various countries have different equipment requirements (although most are the same now). Long haul routes may very well translate to non domestic routes, which means the plane may need to be outfitted with extra equipment that is capable of working on foreign systems. 

There may be some paperwork involved around certifications as well. For example the FCC does not require domestically operated aircraft (commercial and GA) to have radio station licenses; however, it's often required by many foreign entities. A domestic only plane may need to obtain one of these licenses before heading to is destination.     

On a bit of a side note the pilot must also be familiar with the destinations' regulations and methods of operation. For example here is a nice briefing from the FAA on flying into China.