Trapped ions are among the most promising candidates for performing quantum information processing tasks. Recently, it was demonstrated how the properties of geometric phases can be used to implement an entangling two-qubit phase gate with significantly reduced operation time while having a built-in resistance against certain types of errors [M. Palmero et al., Phys. Rev. A 95, 022328 (2017)]. We investigate the influence of both quantum and thermal fluctuations on the geometric phase in the Markov regime. We show that additional environmentally induced phases as well as a loss of coherence result from the nonunitary evolution, even at zero temperature. We connect these effects to the associated dynamical and geometrical phases. This suggests a strategy to compensate for the detrimental environmental influences and restore some of the properties of the ideal implementation. Our main result is a strategy for zero temperature to construct forces for the geometric phase gate which compensate for the dissipative effects and leave the produced phase as well as the final motional state identical to the isolated case. We show that the same strategy also helps at finite temperatures. Furthermore, we examine the effects of dissipation on the fidelity and the robustness of a two-qubit phase gate against certain error types.