Not much. Forth is from an era when it was hard to program small microcontrollers and the tools for doing so were expensive. Today, everybody cross-compiles to microcontrollers.
I've done robotics programming in Forth. I do not expect to use it again.
My understanding is that the link between Arduino and Processing is solely in the IDE. I don't think the two share any common ground beyond just the fact that Java (which Processing is a very thin layer on top of) is similar to C (which Arduino is a very thin layer on top of.)
I've done robotics programming in Forth. I do not expect to use it again.