Symbolism, Imagery, Allegory
Compared to money and weapons, the drum is really a much smaller, goofier symbol in Major Barbara. Dolly came to play the drums at the Army because he wanted to cozy up to Barbara, and it becomes clear pretty quickly that he has zero actual interest in the Army (and probably the drums, too); he's just using both to gain favor with her. So, references to his drum playing are more comic than symbolic of any kind of religious zeal on his part . . .
Later, when he goes to play the drums at an Army event after Barbara has quit, Shaw's stage directions describe him as "mounting the form in an ecstasy of mischief" (2.423)—that is, getting super jacked up—but, again, without any particular genuine zeal for the cause, and more for comic effect.
So, you might just think the drum is a symbol of Dolly's own empty enthusiasm or ridiculousness. But when he and Barbara share a truly tender moment late in the play—when Barbara says she has no intention of abandoning Dolly even though he's going to her father's firm—he's so happy that he exclaims "Oh for my drum!" (3.413).
So, what's the upshot? The drum just seems to be a symbol of Dolly's unbridled enthusiasm and good cheer pretty much regardless of the circumstances—and of his affection for Barbara, too, who probably makes his heart beat just like his drum.