SpiderMonkey dev here. As others have mentioned, Cranelift is one component that's being written in Rust. Eventually we want to use Cranelift as compiler backend not just for WebAssembly but also for JS. After that it might be feasible to port more of the JS JIT to Rust. It might also make sense to use Rust for the parser or regular expression engine, who knows.

There will probably always be C++ code in Gecko, but I firmly believe that writing more components in Rust instead of C++ will (in general) improve security and developer productivity.

It still amazes me that we're actually shipping a browser with a CSS engine (and soon graphics engine!) written in Rust. Even more amazing is that these components are mostly shared with an entirely different browser engine.