Just wanted to know if you can pipe buttercream on top of fondant and around the base of fondant?

Yes, I do it all the time (but I am no expert!)

If I do this the night before the party, will the buttercream dry out too much? Or should I pipe it that day?

Thanks for the input...

It should be ok. It will crust but it shoudln't "dry" out.

I do this all the time, no problems.

Thanks...Since you do not refrig the fondant, will the buttercream be ok to sit out over night? Do I need to cover the cake over night?

Buttercream will be fine to sit out overnight under fondant. I've had bad luck with covering a cake air tight once it's covered in fondant because I find after a day or two the fondant can go a bit gummy. I'd maybe cover it, but not airtight.

Will the buttercream on top of the fondant be ok to sit out over night?

Yes, you can pipe it onto the fondant, and yes, sitting out that night will be fine. Too much covering can make it sweat, I've learned, depending on where you live.