In one place in the yard, there is a carpet of ivy under a black walnut. Would it harm the tree in any way to let the ivy up to the base of the branches? I would like to do this so that the whole scene is green.
No, as long as it doesn't compete for light, the tree is healthy, and the soil situation is normal, ivy is fine on tree trunks. Ivy is not a parasite. Here's a good article.
Walnut and butternut trees try to ensure that there are no competitors for light and water by excreting a substance called juglone. It inhibits the growth of some plants. I cannot find ivy listed as being tolerant which would indicate that it's growth would be stunted.
You could try Virginia Creeper which is tolerant however a well established plant could compete with the walnut tree and inhibit it's growth.
With the price of black walnut wood you could be better off pruning it so it has a good form and then selling it when you want to retire.