I'm working on a quilt using solid black fabric which I've found gets very shiny when ironed. Is there a way to keep the fabric from getting so shiny? Will the shine disappear if I wash the quilt when it's finished?

iron on the "wrong" side
sometimes the shine will wash away

Use one of those teflon ironing shoes. I use them on all my irons all the time!

yes it should go back to normal when it's washed, and it shouldn't happen if you iron on the backside
I use a lot of sateens and it does that too

You can use a pressing cloth.