Are dentures required after removing many teeth?

Not necessarily. You don't have to have dentures. You may want them for appearance and to allow a full range diet. If you are ok with the old grandpa look and enjoy soft foods...Then you can let it ride.
Not required. Most people won't leave their house without teeth in their mouth (natural or false), but it may not matter to you. The few teeth you have left will bear the brunt of your chewing, and will most likely wear out sooner than later. Implants are an option to dentures.
One option. Replacing missing teeth is critical for optimal health. Lost teeth allow an imbalance to occur, which affects not only the adjacent teeth, but also the supporting muscles and joints. The bite is affected and this may cause an overload on the remaining teeth, etc. On down the line with more problems. Best to replace those lost teeth with implants, or removable partial dentures, or fixed bridges.