No they won't kill a tree. I wouldn't leave them in if you are worried about the trees however. You will have a hard time getting them out in the future if left in.

I would recommend starting to save your money and get a set of climbing sticks. I used screw in steps for years but there is no comparison to a set of sticks. You can set a hang on stand with them as quick as you can get up a tree with a climber. They really are amazing. Best investment I have made in years.

A couple things with the steps. Make absolutely SURE they are tight in the tree before you step on them. They don't work very well in Trees with deep bark like an old Cottonwood, they will pull out of bark. If there is frost and they are just metal they will be slick when climbing, be carfeful on cold mornings. Be very careful when climbing them, if you fall or slip and land on one it will be like a gut hook.