Astronomical observations have provided an extensive body of evidence for the existence of disequilibrium chemistry in many exoplanet atmospheres, and this departure from a chemical equilibrium composition may have an impact on the temperature of the atmosphere itself. We have developed a 1D atmosphere model that solves in a self-consistent manner the evolution of temperature and disequilibrium chemistry in the vertical direction. The temperature is solved in radiative-convective equilibrium and the disequilibrium composition is computed including thermochemical kinetics, photochemistry, and vertical mixing. Thermochemical kinetics is based on a reaction network built from scratch that includes 164 gaseous species composed of H, C, N, O, S, Si, P, Ti, He, and Ar, connected by 2352 forward reactions. To investigate the mutual influence between disequilibrium chemistry and temperature in exoplanet atmospheres, we have applied our model to the well-known gas giant exoplanets WASP-33b, HD 209458b, HD 189733b, GJ 436b, and GJ 1214b, which cover different degrees of insolation and metallicity, and to secondary atmospheres that exoplanets characterized in the future may plausibly have. We find that for irradiated gas giants with solar or supersolar metallicity, the corrections to the temperature due to disequilibrium chemistry are relatively small, on the order of 100 K at most, in agreement with previous studies. Although the atmospheric composition of some of these planets deviates significantly from chemical equilibrium, the impact on the temperature is moderate because the abundances of the main atmospheric species that provide opacity, such as H2O, CO2, CO, and/or CH4, are not seriously modified by disequilibrium chemistry. An impact on the temperature greater than 100 K appears in hot Jupiters due to TiO, which is predicted to be seriously depleted by UV photons in the upper layers. However, the extent of this depletion, and thus of its impact on the temperature, is uncertain due to the lack of knowledge about TiO photodestruction. In secondary atmospheres, the impact of disequilibrium chemistry on the temperature depends on the composition. In atmospheres dominated by H2O and/or CO2 the temperature is not affected to an important extent. However, reducing atmospheres dominated by CH4 and oxidizing atmospheres dominated by O2 see their temperature being seriously affected due to the important processing of the atmospheric composition induced by disequilibrium chemistry.