Ask Mr. Smarty Plants

Your ground cover ivy (you don't mention the kind) needs three things to survive; light, water, and nutrients. Your big maple tree needs the same three things; thus they are competitors. The maple tree has the advantage because of a massive root sytem to absorb water and nutrients, and because its trunk and branches can put its leaves up high in order to absorb sunlight. The ivy plant probably can get sufficient moisture and nutrients with its root system, but it most likely isn't getting enough light under the shade of the maple tree. To get more light, it needs a new strategy; grow up the big trunk and branches of the maple tree. This is why your ground cover is becoming aerial.

Mr. Smarty Plants thinks the the answer to your first two questions is no. Generally, ivy vines won't kill trees, but  their presence on the tree is not with out some consequences. If the growth of the ivy becomes excessive, the ivy leaves may block sunlight from the maple leaves, and the weight of the ivy may weaken the limbs of the maple and cause them to break. Removing the ivy won't kill the tree. Just do it carefully so you don't damage the tree bark.

Remove? Don't remove? This a pesonal preference. These two links gives several pros and cons regarding ivy vines in trees.

Trees Atlanta

Classy Groundcovers