I think that carbon and thickfilm (metal oxide) resistors have serious shot noise, and metallic resistors have very little, at least until they get very small. I don't know if thickfilms approach or exceed Poisson-level noise, or exactly what "very small" is; clarifications would be appreciated.
Of course, if you keep DC away from critical resistors, the shot noise becomes a tiny signal gain modulation that can hardly ever matter.
John