Is there any relationship between heartbeat rate and life span of an animal?
Interestingly there is a inverse negative correlation between heart rate and life span, meaning the faster your heart rate is, the shorter is your lifespan. See this figure (from the paper 2 cited below):



When the authors plotted the approximately total heartbeats vs. the lifespan, the amount of total heartbeats was in a pretty narrow corridor:



So it seems that at least the hearts in mammals have a maximum number of strokes they could do. The obvious question what causes this phenomenon is not really answered.
Since the metabolism of small animals is (compared to their weight) higher and also their oxygen consumption is higher because of that, it is hypothesized that this causes more reactive oxygen species and related damage which subsequently leads to an earlier death.

See the references for more details:


Rest heart rate and life expectancy.
Heart rate, lifespan, and mortality risk
Oxidative damage to DNA: relation to species metabolic rate and life
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