If it's a small residentail boiler (burner) as you describe, I would consider the class 2 TT terminals on the fireeye as an operating control only and would not consider it for safety shutdown. I would loop the 20 Amp power feed through all of the safety shutdwon switches, firematic ...
A large boiler burner with a 3 phase blower on the burner, as described in the post above, the safety and operating controls all loop in series to two terminals (In the burner control cabinet) typically 3 and 4.
In either case you would have to read and follow the manufacturer's installation documents where this will be instructed, how to. Should be clear on the residential burner instructions.
On the commercial type with multiple accessories, gas train, combustion air, flue draft fans, all those pieces are just that, seperate pieces each with their own instructions, but now no whole assembly diagram or instructions, no sequence of operation for the assembly. In that case you have to piece together your own assembled package wiring diagram, sequence of operation. It's easy to get instructions for the individual pieces, but I've never seen one with instructions for the enitre assembled package.