As a commissioned boiler inspector, I suggest the following:
-Consider employing a qualified specialist to evaluate your boiler combustion controls and burners. Yes, often these guys seem like they want to sell you something, but just as often they will pinpoint the problem.
-Your boiler will gradually soot up over time and the stack gas temps will increase as a result of it. If the rate of fouling is abnormal for your boiler, then and only then do you have a problem. Your type boiler requires routine cleaning of the firesides to remove soot. If the fouling is rather rapid, then yes, you are correct to suspect that the burner is at fault.
-Also consider that the other half of the fuel/air mix could be the problem. Investigate to see if you are starving your boiler for air. Even tho the burner registers, dampers and blower components are working properly, there might be times where the boiler room gets insuff air flow. A boiler can quickly evacuate the building or equipment room of air and end up in a negative pressure causing the boiler to soot up. If you are in the habit of closing off the windows/doors at night or during bad weather, then the combustion fuel/air mix will go fuel-rich on you if the building/room isn't properly vented.
Picture this: Your boiler is located in an equipment room that is normally sealed off from casual access....You go there to service the boiler, so you prop open the doors to allow full access for people and equipment. You check and adjust the burner combustion settings....Then you pack up and shut the boiler room door....And in doing so, completely change the fuel/air ratio!
-Examine how your boiler operates...how often does it cycle off? If a fully modulating boiler, how often does it idle at low fire or struggle at high fire? Frequent cycling or prolonged firing at either extreme indicate an improperly sized boiler or an improperly managed load.
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It's great that you are monitoring the stack gas temps and then investigating the cause of the increase....Go one step farther. Once you have corrected the present problem, start yourself a graph. Use that graph to plot all your boiler readings over time. Look for trends. Trends will indicate possible problems. Trends are hard to detect unless they are graphed. So record and plot out fuel consumption, make up water consumption, stack temps and anything else you have indications for. Those graphs will also prove helpful and indicate your progress as you make repairs and other changes.