My whole job is boiler control with PLC's. PLC's are used all the time to monitor the boiler status, provide lead/lag control, PID firing rate controls, alarm annunciation, and many additional features that the standard burner control can't provide. Some of the more sophisticated burner controls can implement some of the same lead/lag and firing rate control as a PLC through native serial protocols.
Standard burner controls (called flame safeguards), such as the Honeywell RM7800/40 series and Fireye E-series provide sequencing through the ignition process and are not able to be manipulated through serial communication. These controls time a prepurge, either open damper or closed damper, ignite and prove a pilot flame, ignite and prove a main flame, and monitor the flame in the run state. Any interruption or problem results in a unique fault code.
The serial communications available only allow a few writes: command the boiler OFF (default without comms is ON), command the boiler ON at low fire, or command the boiler on at high fire. Any other firing rate control for a modulating burner with these controls needs to be done with an analog 0-135ohm or 4-20mA signal, depending on the modulating motor used. Older Honeywell mod motors commonly accepted a 4-20mA signal through a 66.5 ohm resistor and a 237 ohm resistor that converted the signal, but the newest revision doesn't work with these. A 4-20mA mod motor can also be used. Many values can be read, including all current status, fault codes, and complete fault history. Most of my experience is with the Honeywell FSG. Also, the S7810M is no longer necessary for Modbus, as the newest S7800A KDM (keyboard display module) has integrated Modbus. You used to have to choose between display and Modbus, and would often go on service calls only to find the Modbus module removed and a KDM in it's place. Modbus implementation with either device is pretty standard and uses 4x registers only.
Advanced FSG's also provide what is known as parallel positioning, or linkageless control. Parallel positioning allows a characterized fuel curve that is commissioned to maximize efficieny, through programmed servos and/or VFD's connected individually to each fuel valve, the air mixture, and the flue gas recirculation. These units also connect together to provide sequencing when desired. These units may allow direct firing rate control through serial communications in addition to start/stop control. Manufacturers include Autoflame, Fireye, Honeywell, and Siemens.
Cleaver-Brooks also has provided their own flame safeguard/parallel positioning systems using SLC500 PLC's programmed through a touch screen to provide the control. This is not common since the UL process for this is very rigid and the liability high. This would normally only be done by a manufacturer that did not want to purchase one of the proven systems, for whatever reason. My company has made UL listed flame control panels, which differ in that they still use another listed flame safeguard for the actual burner controls, otherwise they are very similar to regular burner control panels or any other UL508 panels. We can use PLC's for any logic or relay replacement as necessary in these panels as long as none of the safety features are bypassed or it fails safe. All of the safeties and outputs need to be directly connected to their destination on the FSG without interposing logic.
To sum this all up, PLC's are used all the time in boiler control, but usually in the same way any control contractor would run a boiler - start/stop, firing rate, run status, and fail status.