Established trees can shrug this off without a problem.

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

Just don't pile the mulch, no matter what kind, right up next to the trunk. Spread it as far from the tree as possible. Remember, mulch is supposed to benefit the rhizosphere (root and soil systems interrelationships).

Watch out for sprouting willows too. Even chopped up willow has incredible powers of regeneration.

Can the willow really grow back from the shavings? The last thing I want is willow roots sprouting up because of these wood shavings!!

You won't get a massive forest of treelets but bits of twig and bark can root. It would be worth looking over the area every so often just in case. They'll be easy to pull if you look regularly until they start to decompose. It's not a reason not to use them but just keep an eye open.

Thanks. I'm just really paranoid about willow roots. The trees original roots grew into the old clay pipes of the septic and caused a back flow of water in my basement. I have these grindings close to the house so I don't want roots to grow and bust in my basement wall!

Willows are quite easy to root and willow bark has been used as a rooting stimulant for many other plant cuttings, but generally that has been in a quite wet environment, much wetter then a mulch would be.
I have used willow chips for mulch many times over the years and have not seen any new trees growing from rootings from the chips although there have been some from seeds.

Here is a link that might be useful: willow for rooting