Does a Pokemon "lose" Pokerus while in the Pelago?

Poképelago is essentially just another way of viewing the pokémon in your PC. That is, there's no way to have a pokémon both be in your PC and not be in poképelago. Therefore, it has no effect on Pokérus.

Since placing a Pokémon in the PC puts its ‘time frame’ for contagiousness on pause, as soon as you take it out of the PC it will resume being contagious unless it’s time is up. It does not matter if it is in Poké Pelago, it has no effect on the virus whatsoever. If it does have Pokerus, it should continue having it when you resume the next day. I have a Granbull I found wth Pokerus, and putting it in Poké Pelago did not affect it. Make sure to remember, the longer it is in your party, the less time you have until it is cured.