Given, for example, a birth death process with a set of jump times. These jump times have to be exponentially distributed in order for this process to satisfy the Markov property. Why is this? Why does it have to be an exponential distribution? I know that events occuring at random times are best described using an exponential distribution - but I do not see why this distribution is required.
It isn't . See Renewal Theory. If you are talking about the time of the next event $T_{N+1}$ given the time of the last event $T_N$, then the Markov Property is that $P(T_{N+1}|T_N,T_{N-1}...)=P(T_{N+1}|T_{N})$. Nothing precludes the interarrival times from being any distribution, provided that inter-arriveal times are iid.
Note also that the markov property applies to a much broader class of stochastic models than just the Poisson process.