Does repeated episodes of svt cause heart damage or failure?

Yes: it can causes the heart muscle weaken due to SVT if recurrent and stay in longer duration. it is usually benign but still risk of developing tachycardia related Cardiomyopathy.

No damage if brief: SVT or supraventricular tachycardia is an abnormally fast heart beat that originates from the upper chambers of the heart. If the rapid heart beats are short-lived i.e. minutes or hours, and then the heart returns to normal there will be no damage. If the heart races for a long time i.e. weeks or longer, then there can be weakening of the heart pumping function.