I struggle with fire stuff so take my response as conversation fodder rather than any kind of definitive advice. I feel that the crux of this is the philosophical difference between:
1) The anticipated fire load which which informs regulation on the use of combustible materials. There are just some situations in which you do not want your building made from kindling such that it would burst into a raging inferno in short order. This is about available fuel for the fire.
2) Measures taken to slow the spread of fire and delay the structural damage caused by it with the intent of buying time for occupants to escape. This includes all of the stuff in our fire rated assemblies as well as, I think, sprinklers (flashover etc).
Consequently, I feel that even a sprinklered wood beam contributes to the fire load since, ultimately, sprinklers are not to be relied upon to actually put fires out wholesale in most situations. They are primarily just measures taken to buy time. This leads me to suspect that you cannot use combustible structural elements within structures that are code mandated to be non-combustible.
Rationally, if 95% of your building material mass is non-combustible, does the addition of a handful of combustible beams really add anything meaningful to the fire load? Surely not. However, you may have to seek some manner of building official dispensation to make such a setup street legal. And, usually, that kind of hassle is best avoided.
As an example that is, admittedly, a bit of a reach, check out the system proposed in this paper:
Fire Protection of Windows Using Sprinklers. There, the sprinklers are used to make a fire rated assembly out of a material (glass) not normally suited to that function. But, again, this has not changed the
fire load. It's still just about buying time for folks to high tail it out the door.