Firstly, thanks Jane for asking the question and it is a subject we encountered often but I am afraid it does not have a single definitive yes or no.
Firstly, Ivy (Hedera helix) is not a parasite and it does not actually live off of the trees resources but merely uses the tree as a support.
As Ivy grows it anchors itself using aerial roots and will grow high into the canopy seeking light. Contrary to popular belief Ivy it is not restricting the trees growth at all and will not constrict the trunk or branches.
As far as a woodland tree is concerned I would not worry at all as the Ivy is providing habitat and feeding for a variety of birds and of course the Grey Squirrel.
If you were to have Ivy growing into an ornamental tree then visually it is a problem and if left to run wild will restrict the light to the trees leaves and cause growth irregularities.
Ivy can also be a bit tricky to kill. Sulphate of Ammonia (lawn sand) will work and it pays to cut through the stem with a saw. Make sure you remove at least an inch (25mm) of stem otherwise the plant will 'weld' itself together again.
If you have a gardening or landscape question you need answering then email me at email@example.com
I get a lot of hits on this post from the United States so it seems appropriate that I have come across the Plant Conservation Alliance site that has an extensive write up about Ivy and the ways to control it.