How Are Botox and Fillers Processed out of the Body?

As the other panel members described, both products are very safe and naturally degrade in our bodies with time. New research has demonstrated that our bodies may make new collagen in the areas where the cosmetic dermal fillers begin to slowly break down and degrade - a good thing!
See the below link for further information.

I am assuming you are asking about NON-permanent fillers (such as collagen-based fillers, hyaluronic based fillers such as Restylane and juvederm).
Neither of them persists indefinitely. Each is broken down and removed much like our own tissues (collagen, hyaluronic acid (in eye gel and joint spaces) is broken down and continuously replaced).
Lastly, neither one has been shown to affect our immune response.

Botox or the botulinum toxin is a neurotoxin that acts on neurons to prevent the release of the chemical that causes muscles to work. Although muscles that have been paralyzed with Botox regain function in 3-4 months, this is due to generation of new nerve fibers that don't contain the Botox. The toxin is metabolized, locally by normal protein degrading enzymes, eventually and the by products reused by the body or excreted.
Restylane is composed of hyaluronic acid which is a normal component of the skin dermis. Therefore, there are proteases or proteins that normally degrade this protein. The metabolism is the same for Restylane as the skins hyluronic acid.