A highly sensitive refractive index sensor is proposed by using the resonant coupling of a Tamm state to a Fabry-Perot resonance in slits periodically pierced in a metal film. This new hybrid resonance exists at the interface between a dielectric Bragg mirror and a subwavelength metal grating. Contrary to a classic Tamm plasmon, it is sensitive to the ambient media refractive index due to its confinement inside the grating slits. The proposed sensor output can be either the reflected intensity or preferably the wavelength, and its sensitivity and figure of merit are numerically investigated with the help of rigorous coupled wave analysis. The sensitivity of the studied sensor is 87 nm/RIU for a refractive index range from 1.25 to 1.38, and, at the expense of the resolution, it can reach up to 160 nm/RIU for a grating duty cycle of 50%. The figure of merit is around ${{7.5}}\;{{\rm{RIU}}^{- 1}}$ with a large measuring range. The index sensor performances and operating resonance wavelength can be modified by adjusting the grating geometric parameters (height and duty cycle). The possibility to achieve wavelength modulation in any spectral range makes the proposed grating Tamm structure an attractive candidate to design refractive index sensors and photonic components in the infrared and terahertz domains.