Cold Air return on 2nd Level

I have a 2 story home which is approx. 70 years old. We have no cold air return on the 2nd level and are in the process of replacing the old furnace with a new one. My question is, is it a necessary to have a cold return on the 2nd level? If so what are the reasons. Also the cold air return on the main level is on the floor very close to the stairs going up.

Ideally, each room would have a return. In order to push air into a room, you need an outlet so it will flow. If you always leave the doors open, it isn't a problem. There's often enough room underneath a closed door, but that isn't always the case, especially if carpet is added, and there's now little to no clearance. This can lead to cold (or hot if a/c) rooms.

If there is not a return path in a room but it has an open register, then that room will tend to pressurize slightly. That will allow part of the air to escape underneath closed doors...or outside through any opening it can find (wall, window, attic.) As was already mentioned it will also reduce flow into the room (higher pressure to overcome) causing temperature differential with other sections of the home.