The effect of doping on the carrier-phonon interaction in wurtzite GaN is investigated by pump-probe reflectivity measurements using 3.1 eV light in near resonance with the fundamental band gap of 3.39 eV. Coherent modulations of the reflectivity due to the E-2 and A(1)(LO) modes, as well as the 2A(1)(LO) overtone are observed. Doping of acceptor and donor atoms enhances the dephasing of the polar A(1)(LO) phonon via coupling with plasmons, with the effect of donors being stronger. Doping also enhances the relative amplitude of the coherent A(1)(LO) phonon with respect to that of the high-frequency E-2 phonon, though it does not affect the relative intensity in Raman spectroscopic measurements. We attribute this enhanced coherent amplitude to the transient depletion field screening (TDFS) excitation mechanism, which, in addition to impulsive stimulated Raman scattering (ISRS), contributes to the generation of coherent polar phonons even for sub-band gap excitation. Because the TDFS mechanism requires photoexcitation of carriers, we argue that the interband transition is made possible at a surface with photon energies below the bulk band gap through the Franz-Keldysh effect.