I didn't replace the washers on my Solstice, and it was fine. Just watch for leaks.
I have had to replace oil drain plug gaskets. Either they just disintegrate over time (plastic/rubber), or someone previous to my ownership decided to see how tight they could install the drain plug and ruined it.
My Subaru has crush washers on the drain plugs (engine, transmission, and I think both differentials but I don't remember for sure). These are replaced at every change. They're about $1 from the Subaru dealer, and if you buy an oil filter there (which is just as cheap as the auto parts chain stores), they just trot one out at the same time.
2006 Solstice, Mysterious 5-speed