Polyester CAN be dyed.
Unlike cotton or wool, dyeing polyester requires the use of disperse dyes. While disperse dyes are non-toxic, they require exceptionally high temperatures, and additional toxic carrier chemicals, to effectively dye synthetic fabrics. For these reasons, experts do not recommend that novices attempt to dye polyester through immersion. Alternatives to immersion dyeing are readily available, though, and may be a viable solution, depending upon your desired results.
Crayon manufacturer, Crayola®, also produces fabric crayons for use on polyester. Most fabric and craft stores carry these, as well as national department stores. You use the crayons to draw on a piece of paper, and then transfer from the paper as an iron-on.
Heat Set Ink
Heat set ink is another option, best for the application of designs, and not full garment dying. You apply this ink to a rubber stamp, stamp the design on paper and transfer via the same method as the crayons. You can purchase heat set ink from most rubber stamp suppliers. Large-scale fabric stamps are also available.
Powdered Disperse Dyes
You can also use basic powdered disperse dye, such as PROsperse®, in transfer applications. Mix the dye into just enough boiling water to dissolve the powder. Dilute this solution with cold water until the desired color is reached, keeping in mind that the intensity will increase once ironed on. Paint or screen this is onto a piece of paper. Once dried, iron it onto the fabric, using pressure. If you are using an iron with holes in it, be mindful to keep the iron moving constantly to prevent hole patterns from appearing in the transfer. You can also use a heat transfer press.
For immersion dyeing, follow the same steps as with powdered dye. Follow the manufacturer's instructions after this step, as each product is different.