What is the sum of all the positive two-digit integers where one of the digits is three times the other?
We have two cases:

$\bullet$  Case 1: The tens digit is three times the unit digit. In this case we have $31,$ $62,$ and $93.$

$\bullet$  Case 2: The unit digit is three times the tens digit. In this case we have $13,$ $26,$ and $39.$

Sum up the two cases: we have $31+62+93+13+26+39 = \boxed{264}.$