IMHO, remove as much as possible. As part of the landscaping work I had done last year, all the shrubs were removed, three small trees cut down, another limbed, all the (fesque) grass and wood chips from previous islands removed, the yard mildly regraded, and a load of topsoil onto which the sod would be placed spread evenly. All that was taken out had to be put into a dumpster.

If you are laying sod and have space to put the removed materials on another part of your property, you can do a lot of prep work fairly inexpensively by hiring out someone with a Bobcat or other frontend loader and getting a truckload or two of topsoil delivered.

You're going to invest quite a bit in sod, regardless of grass type. It will have a much better chance of "taking" as well as look a lot better if you invest extra in proper preparation.