The most convenient spot for my new compost pile happens to be under a young apple tree. I would put it there, but I have a few questions first.
Yes, heavy shade is fine. In fact, it may be better, because the pile will dry out slower.
The compost and its leachates will be good for the tree. I don't know about apples in particular, but the tree will grow roots up into the compost, and you'll need to keep cutting them. On balance, it still should be good for the tree. It's a minor nuisance. Don't let the roots get deep into the compost (say, 2-3 years without turning) and then cut all at once - that will stress the tree.
I don't know if it will promote disease in the fruit. If concerned, you might avoid turning the compost while fruit is ripening, to avoid throwing up lots of mold spores.
Most tree roots are within 6 inches of the surface and extend from the trunk to the drip line because that's where the oxygen and water are. If you raise the existing grade by more than an inch or two over more than two thirds of the rooting area there is a good possibility that the tree will decline as the roots starve for oxygen.
This is most commonly seen in new construction areas where contractors put safety fencing around an established tree to "protect" it and then kill it by raising the grade a foot.
Younger trees have a better chance of growing new roots up into the compost but that's not what you want either.
I do not recommend putting a compost pile under the tree unless it occupies no more than twenty percent of the rooting area.
We had a pear tree above our compost pile (the pile was at the bottom of a small 5 foot dip). Anyway, that tree was amazing. Actually, it's still there. This was at my college in Virginia. The tree just gave tons of fruit and was absolutely amazing. I think we sold $300 worth of pears and that's after the students took a good share.
