I would seriously doubt, even if there is a mark on a tree when you tear down the tarp, that it does any lasting damage to any tree.
Trees are pretty tough plants, and if the bark isn't removed all around, or perforated to the core, the tree is very unlikley to suffer any ill-effect.
If you are worried a couple of sticks pushed down behind the ridgeline, before you tension it should see that the load is spread enough to avoid any problems.
cheers
Gareth
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