We present a high-resolution set of adiabatic binary galaxy cluster merger simulations using FLASH. These are the highest resolution simulations to date of such mergers using an adaptive mesh refinement grid-based code with Eulerian hydrodynamics. In this first paper in a series, we investigate the effects of merging on the entropy of the hot intracluster gas, specifically with regard to the ability of merging to heat and disrupt cluster 'cool cores.' We find, in line with recent works, that the effect of fluid instabilities that are well resolved in grid-based codes is to significantly mix the gases of the two clusters and to significantly increase the entropy of the gas of the final merger remnant. This result is characteristic of mergers over a range of initial mass ratio and impact parameter. In line with this, we find that the kinetic energy associated with random motions is higher in our merger remnants which have high-entropy floors, indicating that the motions have efficiently mixed the gas and heated the cluster core with gas of initially high entropy. We examine the implications of this result for the maintenance of high-entropy floors in the centers of galaxy clusters and the derivation of the properties of dark matter from the thermal properties of the X-ray-emitting gas.