On Shabbos, is it permitted to turn on the water if there is a sponge lying in the sink (or is it considered “laundering”)?
Is it permissible to remove the sponge from the sink even though it is muktzah (since one needs the makom)?
It is permissible to turn on the water in the sink. The sponge should be removed indirectly, by gently lifting it up with a plate or the like.
There is no problem of laundering being that it is already soaked with water, and in addition the sponge is made of synthetic material. It is also not your intent and you have no gain from the sponge being wet.
Because of the concern of muktzeh it should moved indirectly.