"I have a shirt that is made of rayon. What is rayon anyways, and how can I dye it?"
Answer: Rayon (viscose) is a processed cellulose fiber, and can be dyed with fiber reactive dyes, just like natural cellulose fibers.
Fiber reactive dye is the kind that you find in tie dye kits, because t-shirts are typically 100% or mostly cotton, and these are a popular canvas for tie-dye.
The fiber reactive dye creates a molecular bond with the cellulose material, so that the dye actually becomes part of the fabric. This means the dye will remain vibrant and permanent throughout the life of the clothing (more about fiber reactive dye and where to find it).
Fiber reactive dyes work in cold water, so any tutorial on fiber reactive dye shouldn't involve heat. It is also advised to soak your fabric in a mordant (thing that makes the dye stick better). For fiber reactive dyes, soda ash works great to help the dye bond to the fiber molecules.