Empirical studies suggest that contact patterns follow heterogeneous inter-event times, meaning that intervals of high activity are followed by periods of inactivity. Combined with birth and death of individuals, these temporal constraints affect the spread of infections in a non-trivial way and are dependent on the particular contact dynamics. We propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event times, and leave and enter the system at fixed rates. We study how these temporal properties affect the prevalence of an infection and estimate R0, the number of secondary infections, by modeling simulated infections (SIR, SI and SIS) co-evolving with the network structure. We find that heterogeneous contact patterns cause earlier and larger epidemics on the SIR model in comparison to homogeneous scenarios. In case of SI and SIS, the epidemics is faster in the early stages (up to 90% of prevalence) followed by a slowdown in the asymptotic limit in case of heterogeneous patterns. In the presence of birth and death, heterogeneous patterns always cause higher prevalence in comparison to homogeneous scenarios with same average inter-event times. Our results suggest that R0 may be underestimated if temporal heterogeneities are not taken into account in the modeling of epidemics.