No, time dilation does not cause a clock to run faster than its stationary counterpart. Time dilation always results in a clock running slower compared to a clock in a reference frame where time is not dilated.
Time dilation arises from the fundamental principles of relativity, both special and general relativity. According to special relativity, as an object's velocity increases relative to an observer, time appears to slow down for that object from the observer's perspective. This effect is commonly referred to as velocity time dilation.
In general relativity, the presence of a strong gravitational field causes time to move slower near massive objects. Clocks located closer to a massive object experience a greater gravitational time dilation and tick slower compared to clocks located further away.
In both cases, whether it is velocity time dilation or gravitational time dilation, time always appears to run slower for the moving or gravitationally influenced clock relative to a stationary clock.
Therefore, time dilation never causes a clock to run faster than its stationary counterpart. It can only result in a clock running slower compared to a clock in a reference frame where time is not dilated.