No cite, but trust me. Nothing grows after death. And Shagnasty is absolutely right about the withering which makes them look longer.
The “lengthening” of fingernails and toenails doesn’t begin to become apparent until the body is in a state of early decomposition. I have examined God knows how many freshly dead bodies and the fingernails and toenails, just like the face, remain exactly as they would look in life, to the point that I can diagnose maladies by abnormalities of the fingernails. (Trace clubbing, subungual hematoma…) But when the guts start to putrefy and the fingers start to mummify, the drying flesh pulls back from the fingernails, exposing more of them, making them look long.
I can see how this got started. When I look at a corpse in early decomposition (or advanced) with some mummification of the digits, I am often startled by the apparent length of the nails, and think something like “wow, those’re really long”. Then I catch myself, and remind myself that they are mummified. The nails wouldn’t have looked that long in life.
You don’t tan after death because, to tan, your melanocytes have to produce melanin and deposit it at the dermal-epidermal junction, or the line between the transparent outer layer of your skin and the opaque underlying layer. (Same layer as where tattoo ink resides, roughly.) Producing melanin requires active synthesis by the nuclei and endoplasmic reticulum of the cell, and only active cells, breathing in oxygen and using up sugar, can synthesize. Once they die, the silence of fallen snow descends over everything. The only changes in the skin from then on will be due to rot.