Does Ivy kill trees

If you want the ivy off, just leave it alone to die back on its own. Be really cautious about anything that exposes the tree where the bark is off - bark removal is more dangerous to the tree than ivy is. When you're removing the ivy, do it very gently - no hearty tugs on the ivy.

Right.
Just cut the ivy's roots near the bottom of the trunk of tree and then wait until next Spring for the ivy to really dry up. You'll be able to remove it then with hardly any damage.
Keep an eye on the ivy trying to regrow, it's relentless.

Not at relentless as bittersweet, but just about.

Joe(Get my Clippers!!)Nation

Is it a good thing to keep off your trees? Or does it really even matter?

Ivy degrades the bark, and can lead to some other problems.

the trees will be healthier without it.

I grow some ivy up trellises to shade my house, but try keep it off of living things...

You want to prevent ivy from getting onto trees to begin with.

Your strategy of cutting it at the roots was good - keep an eye out for suckers trying to start up in a slightly new spot and cut/cut/cut those new suckers before they can start up the tree - as JoeNation has noted, it is relentless.

You won't believe how tall this tree is - and the ivy appears to go pretty much to the top. I was worried it would strangle the tree and then the tree were to die and fall on our house.

I began trying to cut some of the roots, but they were so big and thick that my husband got involved. We were cutting and pulling up roots all over - damn those things were thick and strong.

Then hubby tried grabbing some of the vines to pull them off the tree, he began swinging on them (not on purpose) - he almost went slamming back into the tree trunk. My daughter came out to take pictures - I was really hoping she was filming so I could put on utube when he ended up being sent to hospital with a concussion or something. Fortunately he didn't - unfortunately she wasn't filming, only taking pictures.

The vine is still hanging from the tree - ivy win/hubby lose.

You may need to buy loppers with a good ratchet mechanism to cut through the roots of the ivy. Not a bad thing, as good loppers come in handy if you have large trees and shrubs and are doing the semi-annual maintenance cuts.

Not sure what a lopper is - but we were able to cut the roots - hubby has a heavy duty cutting thingy. He broke the lighter weight one I was using on this mean vine so he went and got the heavy artilary. So we did manage to cut all the roots off the tree. It now just has the vines going up from a level that can't be reached other than the mean one hubby was swinging on.