The rate constants for the formation, destruction, and collisional excitation of SH$^+$ are calculated from quantum mechanical approaches using two new SH$_2^+$ potential energy surfaces (PESs) of $^4A$ and $^2A$ electronic symmetry. The PESs were developed to describe all adiabatic states correlating to the SH$^+$ ($^3Sigma^-$) + H($^2S$) channel. The formation of SH$^+$ through the S$^+$ + H$_2$ reaction is endothermic by $approx$ 9860 K, and requires at least two vibrational quanta on the H$_2$ molecule to yield significant reactivity. Quasi-classical calculations of the total formation rate constant for H$_2$($v=2$) are in very good agreement with the quantum results above 100K. Further quasi-classical calculations are then performed for $v=3$, 4, and 5 to cover all vibrationally excited H$_2$ levels significantly populated in dense photodissociation regions (PDR). The new calculated formation and destruction rate constants are two to six times larger than the previous ones and have been introduced in the Meudon PDR code to simulate the physical and illuminating conditions in the Orion bar prototypical PDR. New astrochemical models based on the new molecular data produce four times larger SH$^+$ column densities, in agreement with those inferred from recent ALMA observations of the Orion bar.
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