While the practice was started in either 1240 or 1241 by King Henry the Third, either due to the popular fashion at the time of having whitewashed walls, or potentially due to it's claims for keeping buildings strong and clean, it only lasted a little over four hundred years.

Between 1666 and 1680, Christopher Wren, the prominent architect, was tasked with rebuilding much of London after the Great Fire. One of his acts was to remove the whitewashing of the White Tower, no longer popular due to new advances in building materials. He also re-did all of the windows, re-built the original staircase leading up to the main entrance, and carried out work on the Horse Armoury.

So, in short: They stopped whitewashing it because Christopher Wren didn't think it was cool, and it never came back in fashion enough, or was used often enough as a Royal residence, to consider re-doing it.

Sources:

The Annals of London: A Year-by-year Record of a Thousand Years of History, by John Richardson

London, by Rob Humphreys and Judith Bamber

https://www.assetsure.com/homes/listed-buildings-insurance/tower-of-london/