Relativistic Effects Observed with Time Travel Disclaimer: My domain knowledge on these topics is pretty minimal. I'm a "physics fan".
From what I understand about relativity, if there are two identical objects, A and B, and A is stationary and B is not, then A will observe that B's time is elapsing more slowly than its own. It will also notice that B is shorter in the direction of travel than A is.
Now let's assume that A and B could somehow see through time as well as space (this is potentially where the question breaks down). I suppose they would see something like a series of cubes, representing each instance in time, with all the objects in different positions.
If both A and B were now stationary in space, but B was traveling twice as fast through time than A was: Would there be any relativistic effects A would see as it is observing B?
 A: 
Then A will observe that B's time is elapsing more slowly than its
  own. It will also notice that B is shorter in the direction of travel
  than A is.

But, it is also true that B will observe that A's time is elapsing more slowly than its own and B will also notice that A is shorter in the direction of travel than B is.
This is because motion is relative.  We can't really say that A is stationary since, according to B, it is A that is moving.
What we can say is that there are two identical objects in relative motion with respect to each other.


If both A and B were now stationary in space, but B was traveling
  twice as fast through time than A was. Would there be any relativistic
  effects A would see as it is observing B?

In special relativity, if both A and B are inertial and at rest with respect to each other, their clocks run at the same rate so that's that.
Now, if A and B are Rindler observers, both A and B agree that their spatial separation is constant but each experiences a different constant proper acceleration.  Thus, their clocks will run at different rates.
However, in an inertial frame of reference, the spatial distance between A & B is changing so not everyone agrees that A & B are relatively at rest.
This is all to point out that there is no simple setup with a straightforward answer to your question.
