(Two answers included here)
My first thought was
Iron
I can make you dizzy
Lack of iron (hemoglobin) in the blood will make you dizzy and lightheaded.
I can ready athletes for games
Blood doping is a common practice, where an athelete sets aside their own red blood cells (iron) and adds them before an event.
I can shave sheep
Shears and clippers were often made of iron. (Steel is more common now, but steel is made from iron.)
I can show you images on screens
In old CRT (Cathode Ray Tube) style monitors, an (iron) electron gun would shoot electrons at the screen to create the image.
And with my help... you will see the truth when things go wrong
When you get lost and use a compass, the magnetized iron will let you know which way is which.
I can even do a mixture of all these things at the same time
This one is a little vague, but I suppose an athelete could be dizzy from lack of hemoglobin while lost on his way to a blood doping session before his televised event. I don't know. :)
And yet despite all of these things which I can do
I do not have the heart or soul to be able to move anyone or anything
Iron is inanimate. Also, it is the element with the highest binding energy per nucleon. Elements with nucleii lighter than iron can be fused to release energy, elements heavier than iron can release energy when their nucleii are split. (It does not contribute energy in fusion or fission reactions).
Then someone told me the answer was wrong. So here is my second attempt:
Shear
I can make you dizzy
Wind shear can make you dizzy
I can ready athletes for games
Swimmers shave before an event
I can shave sheep
You shave sheep with shears.
I can show you images on screens
LCD monitors take advantage of shear...
And with my help... you will see the truth when things go wrong
Umm....
I can be represented by a film
The Matrix: Shear can be represented by a shear matrix.