Not unless it's sitting in a puddle. As long as the soil is draining properly it should be fine.

How long after planting will the shoots come up? I planted about 2 inches deep, bare root has 6 eyes and in an area with at least 6 hours of sun.

In zone 8 2 inches is a bit deep if that's how much is covering the eyes. You might want to gently brush away some soil until the eyes are visible. You can add a bit of dirt so they're just covered but you don't want them any deeper.

There are a couple of tiny shoots today. Is there anything else need to be done to ensure the young plant health?

If you mulch your peonies you'll want to move it away from the new shoots and from above the root entirely but that's it. If you plan to fertilize now's the time, too, before the shoots get too big. Otherwise just stand back and watch them grow!

How often do young peonies require fertilizer and what is the best time?