You are a:
I can burn, I can attack,
I can ache, I can break,
Heartburn, heart attack, heartache, heartbreak.
Add my head, you might find me in a shack,
Add another 'H' (its 'head', or first letter) to get HEARTH, the floor of a fireplace (which might be used to warm a shack).
Move my head, you can go on a long trek.
Move the 'H' to the end instead to get EARTH, a planet you can travel for thousands of miles...
I can stop, I can eclipse, I can throb, or be warming,
You can stop a heart, have a Total Eclipse of the Heart (if you're Bonnie Tyler), be a heartthrob, or be heart-warming.
Remove my tail, then use those ears,
Remove the tail (i.e. last letter) and you can HEAR.
Remove my legs, its so scorching!
Remove the penultimate letter ('legs'), and you have HEAT.
As for the title:
We've really got to the HEART of the matter when we realise what a vital organ it is!