Why does moving through time not require energy? Moving through the other three dimensions necessitates energy. But why doesn't moving through time necessitate energy?
 A: Moving through the other three dimensions necessitates energy. But why doesn't moving through time necessitate energy?
Like OrangeDog and peta said, it doesn't take any energy to move through space. The Earth is moving through space, but it isn't consuming any energy. And like what ACuriousMind said, moving through time doesn't make much sense. To be blunt, it's just a figure of speech that people have grown overly accustomed to. You don't really move through time. When you suffer time dilation it's like you're in slow-motion mode, that's all. An extreme example of this is offered by the stasis box. Like time travel, it's science fiction, but IMHO it gets to the heart of the matter. No motion of any kind occurs in the stasis box. Light doesn't move, electrochemical signals don't move, nothing moves. So when I shut you inside the stasis box for five years then open the door, you think I opened it immediately. You "travelled to the future" by not moving at all while everything else did. 
A: Moving through space at a uniform pace does not require energy, or force (Newton's 1. law), but accelerating through space does (Newton's 2. law). Similarly, moving through time at a uniform pace does not require a force, but if you're accelerating, your time will change wrt. a non-accelerating observer, so in a way you might say that you accelerate through time.
For instance, if you throw yourself in a black hole, free-falling toward the horizon, your time will pass "normally". But when your realize your mistake and use your jetpack to accelerate up of the potential well, back to civilization, you will find that what took ten minutes for you, took 100 years at Earth, so you have increased your speed through time.
A: Velocity is defined as distance over time, so based on that premise, you are moving through time at the rate of 1 hour per hour, or 1 minute per minute, or 1 second per second.  You cannot go faster than 1 hour per hour relative to your own "clock". You are simply experiencing the one-directional "arrow of time" (Sean Carroll), whereas in space you have a choice in which direction to go. If you are moving through space, you inevitably progress through time.
A: I am going to interpret your question as, it takes energy to accelerate, decelerate, and overcome friction in the other 3 dimensions, why does moving through time not require energy?  
The answer is very simple.  In the case of moving with respect to the 3 spatial dimensions - you move. For the case of the time dimension, time "moves" with respect to you!
An example might make this clear. If you sit still and watch a clock, you spend no energy, but you are "moving" though time (time moving through you, would be more accurate).
