Dying relates to death. Dyeing relates to soaking items in colored solutions. Dieing is a misspelling.
A similar distinction applies to the past participles died and dyed. Die becomes died, and dye becomes dyed.
Dyeing and dying are occasionally mixed up—for example:
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]
These are positive examples:
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]