Well you're right, there are mixed opinions about letting ivy grow up trees - I incline to the view (from experience) that it shouldn't be allowed to do so. Depending on the variety of ivy (so called English ivy covers a lot of varieties), and variety of tree, the likelihood is it will end up in the crown of the tree, blocking sunlight and air to the foliage and branches. Eventually, when you look at the tree, all you'll see is what looks like an ivy tree, with little sign of the tree beneath. The ivy roots into the bark only to climb upwards, it doesn't take nutrients from the tree via that route, though it certainly competes for both water and nutrients with the tree roots. Some gardeners believe Ivy only grows up trees which are already partially 'unwell', so to speak, usually because the tree often falls in later years - the likelihood is the weight and density of the ivy has completely smothered the tree and caused or hastened its demise.
It will by now be well rooted into the bark of the tree, so if you decide to remove it, just cut the woody and green stems at the base of the tree, but don't attempt to pull the growths off if there are a lot, there's a risk they may remove the bark if you do. The growths will shrivel over time and all the leaves will drop off. The stems may remain for some time, but should eventually start to disintegrate.