Is it permissible to take a small branch from a neglected olive tree that its owner has not benefited from for more than thirty years?

If custom dictates permission to take a branch and benefit from it, then there is no harm in taking it. That which is permitted by custom is like that which is explicitly permitted by text. And whatever the harvest leaves of crops, and the gleaners leave of dates, due to their disinterest in it, becomes the possession of the one who takes it.