Salbutamol is beta-2 adrenoceptor agonist used to treat the respiratory symptoms of asthma. We learned that it is beta-2 adrenoceptors in the lungs and beta-1 adrenoceptors in the heart. So why does overdose of salbutamol cause a rapid heart rate? 

Activation of beta-2 adrenoceptors in the airways promotes bronchodilation, reduction of airway secretions, and stimulation of mucociliary clearance.  Thus beta-2 adrenoceptor agonists are used in treating the symptoms of asthma. Meanwhile, in the heart, beta-1 adrenoceptor activation has inotropic and chronotropic effects, increasing contractile force and heart rate, respectively.

For the treatment of the symptoms of asthma without causing cardiovascular adverse effects, selective beta-2 adrenoceptor agonists would be the preferred.  Salbutamol is an example of a selective beta-2 adrenoceptor agonist. However, the beta-2 and beta-1 adrenoceptors are very similar, so salbutamol is not entirely selective. Salbutamol shows dose-dependent selectivity for beta-2 adrenoceptors but does still act as a weak beta-1 agonist.  Thus, on overdose, the beta-1 agonist activity of salbutamol can start to cause cardiovascular adverse effects by activating beta-1 adrenoceptors in the heart to increase the force and rate of heart contractions.