I'm assuming the OP is talking about furnaces which are called fired heaters in my line of work. My company supplies fired heaters primarily for oil refineries. While fired heaters come in many configurations, almost all have a radiant and a convection sections.
I'm not sure exactly what you were asking about the "importance" of a convection section. Flue gas temperatures are high in the radiant section (~1500F) which makes radiant heat transfer an efficient means of transferring heat. If you had only a radiant section the flue gas would be leaving at 1500F which is an awful waste of heat. The convection section allows for further heat extraction from the flue gas at lower temperatures. At these lower temperatures, convection is a more effective mechanism of heat transfer.
As far as increasing convection duty. It depends on what options might be available you. If you could lower the inlet temperature of the process stream you would pick up more duty. If you cannot change the process, the convection section could be modified. Additional convection tubes could be added. Convection tubes (at least some of them) usually have fins or studs to increase their heat transfer efficiency. It may be possible to install tubes with more fins.
If you provide a little more information we may be able to give you a better answer.
