john hamilton wrote:
Our local council has a lot of trees with ivy growing on them like this.
They have been notified about it, but they do nothing.

See picture on this link on Tinypic webpage.

http://tinypic.com/r/16hj0iw/5

Would ivy growing like this do any damage to a tree?

Only in the sense that it increases the wind loading in winter and may
result in the thing being blown over in storms. A healthy tree can
tolerate ivy growing on it, but the example you show is pretty extreme.

Regards,
Martin Brown