Dying relates to death. Dyeing relates to soaking items in colored solutions. And a similar distinction applies to the past participles died and dyed. Die becomes died, and dye becomes dyed.
Examples
It’s easy to get dying and dyeing confused. For example, these writers mix them up:
Each part of the dying process—soaking, tying, dying—was given its own station. [Foster's Daily Democrat]
The English philosopher Francis Bacon (1561-1626) compared alchemy to a dyeing man who told his sons “I have left some gold for you buried somewhere in our vineyard” … [Anime Comics and Media]
And these writers use the words well:
She acts as an advocate for the dying and their families by helping make sure their wishes are followed. [Los Angeles Times]
Other displays will feature edible and toxic fungi, mushrooms used for dyeing wool, mushroom cultivation kits, mushroom books, microscopes and fungal ecology. [The Humboldt Beacon]
