Burner tips can become plugged for several reasons. If there are any C3 olefins in the fuel gas, they will polymerize at the burner tip and plug the holes with a carbonaceous substance. Corrosion products can plug the burner tips as well. Typically, the refinery fuel gas is treated with an aqueous amine that saturates the fuel gas with water at temperatures higher than ambient. As the fuel gas is routed to individual heaters the temperature decreases and water will condense. Carbon steel piping will corrode and it is those corrosion products that can plug the burner tips. As Evan indicates, having a coalescer and knock out pot for each branch of the fuel gas system should prevent that moisture from reaching the burners. It is common for the fuel as piping to be constructed of stainless steel downstream of the coalescer. Some installations have the fuel gas lines heat traced and insulated downstream of the knock out pot. There are concerns about corrosion under insulation as the piping does not operate at high enough temperatures to prevent that damage mechanism.
Plugging the fuel gas line is not very common. If the fuel gas has H2S, liquid water and oxygen in contact with one another, a chemical reaction will occur that will form elemental sulfur. It is possible for that deposit to plug large diameter lines.