Unsteady baryonic gamma-ray burst (GRB) jets release variable photospheric emission and can have internal shocks occurring far above the photosphere. We generically formulate a photospheric emission model of GRBs including Compton up-scattered photospheric (UP) emission off the electrons (and positrons) in the internal shocks, and find that the photospheric emission may correspond to the traditional (Band) component at ≲ 1 MeV and the UP emission to the high-energy emission observed by Fermi/LAT for some GRBs at ≳ 10 MeV. The two components can be separate in the spectrum in some cases or can mimic a smooth broad Band spectrum in other cases. We apply our formulation to the well-studied long and short LAT GRBs, GRB 080916C, GRB 090902B, and GRB 090510, and find reasonable parameters for fitting the time-binned spectra. The observed delays of the high-energy emission with respect to the MeV emission which are large compared to the variability times are unlikely to be due to simple kinematic effects of a non-evolving jet. These delays may instead be attributed to the temporal evolution of the physical parameters of the jet, and thus the delay timescales could provide a potential tool for investigating the structures of GRB jets themselves and their progenitors. The difference of the delay timescales of long and short GRBs inferred from the Fermi data might be due to the differences in the progenitors of long and short GRBs.