We present resonant Raman scattering experiments on nanotube samples enriched in metallic armchair single-wall carbon nanotubes (SWCNTs). We establish the transverse optical (ATO) phonon frequency for the (5,5) through (10,10) armchair species, ranging in diameter from 0.68 to 1.36 nm. The frequencies show a strong diameter dependence similar to that previously observed in semiconducting nanotubes. We show that the ATO frequencies in armchair SWCNTs are dramatically upshifted from those of semiconducting SWCNTs. Furthermore, using electrochemical doping, we demonstrated that the ATO frequencies in armchair SWCNTs are independent of the position of the Fermi level. These results suggest that the upshift is a result of a Kohn anomaly involving a forward-scattering mechanism of electrons close to the Fermi level. This is in contrast to the well-known Kohn anomaly that dominates the downshift of the ALO and E2g phonons in nonarmchair metallic SWCNTs and graphene, respectively.