High carbon material such as wood chips laid on the soil as mulches do not "rob" the soil of nutrients. Dr. Alex Shigo, when he was chief of research for the U. S. Forest Service, found that wood chip mulches were necessary in the forest to maintain the fungal dominant environment trees need to grow healthy. Wood chips incorporated into the soil can, temporarily, tie up Nitrogen since the Soil Food Web will concentrate on digesting them instead of feeding the plants. Here is a link that might be useful: wood chip mulches

Remove the shavings if you can, it'll not only tap nitrogen, it'll settle as they decay off. That leads to nasty dips in your lawn. Inevitably, you won't be able to get all the bits of wood out. That's OK, during cooler weather you can fertilize just a little extra (maybe ten to twenty percent) in those locales to keep nitrogen levels up. I generally recommend the use of Milorganite or soybean meal as the nitrogen release is slower, gives the grass a good chance at getting enough, and also supplies plenty for the wood to rot. Since you may still get settling, you can add up to a quarter inch of top soil at a time to those areas to start re-adjusting them. The grass will grow right over it. Repeat once every six weeks until the soil levels out.

All true-I've seen the water-repelling tendency of some mulches sometimes. Camp's suggestion to preset the soil is a good one. I'd like to just quickly address one point here: folks often will say or write things that indicate that organic materials, in the process of being broken down by soil microbes, will "rob" that soil of nitrogen. In actuality, no such thing happens. What does happen is that these microbes will temporarily use or tie up that nitrogen in their processes of breaking stuff down. But then you know what these things do? They die, and all that N is re-released back into the soil matrix. So it's really more of a borrowing. Still important to know, but I just wanted to slightly readjust that paradigm. Soil microorganisms are the single most important factor in tree health, beyond such obvious factors as light and moisture. And those fungi that work with tree roots-mycorrhizae-are true workhorses in both our planted environments and in the large thing we call nature. Indispensable.

If you have the space, an ideal thing would be to heap them up and let decomposition proceed for a year or so. Stump chips do eventually become a very nice, high-organic-matter-content material. So in other words, treat them as compostable material, with or separately from any other compost setup you may have. In fact, again if space permits, I'd likely add them to my regular compost heap. If it's just too much volume-and you don't have the space-probably best to haul them off somehow.