We can rule out thyroid since hGH can lead to a reduction in T4 and T3 levels (which is why it's best to use thyroid medication during extended hGH use ...). hGH can increase serum sodium levels (Graham et al. 2007) which would lead to water retention which can in turn lead to increased blood pressure. The increase in blood pressure might put more strain on the heart because it will have to work harder to "push" the blood through the vascular system (because of the increase in blood pressure, especially when AAS are also used).
The added workload can either be accomplished by having the heart produce a stronger push (stronger myocardial contraction ... the heart is a muscle and if it's strong enough it can contract harder) or more frequent pushes (increased heart rate).
Boguer et al. (1996) have shown that hGH use lead to an increase in cardiac output of 30-40%, mostly because of an increase in nitric oxide production which leads to an increase in vasodilatation.
However this effect (increased blood pressure) is speculative since a lot of studies on hGH tend to show a decrease in blood pressure and some even show a decrease in resting heart rate (Sverrisdottir et al. 2003).
hGH seems to reduce the excretion of adrenaline, which might mean that it makes drugs like clenbuterol, ephedrine and albuterol more effective to... it is worth considering that the heart rate increase comes from a combination of several products, not only from the hGH itself.