We perform the first systematic study of how dynamical stellar tides and general relativistic (GR) effects affect the dynamics and outcomes of binary-single interactions. For this, we have constructed an N-body code that includes tides in the affine approximation, where stars are modeled as self-similar ellipsoidal polytropes, and GR corrections using the commonly used post-Newtonian formalism. Using this numerical formalism, we are able resolve the leading effect from tides and GR across several orders of magnitude in both stellar radius and initial target binary separation. We find that the main effect from tides is the formation of two-body tidal captures that form during the chaotic and resonant evolution of the triple system. The two stars undergoing the capture spiral in and merge. The inclusion of tides can thus lead to an increase in the stellar coalescence rate. We also develop an analytical framework for calculating the cross section of tidal inspirals between any pair of objects with similar mass. From our analytical and numerical estimates, we find that the rate of tidal inspirals relative to collisions increases as the initial semimajor axis of the target binary increases and the radius of the interacting tidal objects decreases. The largest effect is therefore found for triple systems hosting white dwarfs and neutron stars (NSs). In this case, we find the rate of highly eccentric white dwarf - NS mergers to likely be dominated by tidal inspirals. While tidal inspirals occur rarely, we note that they can give rise to a plethora of thermonuclear transients, such as Ca-rich transients.