The question is summed up in the title. What happens exactly?
You cannot completely cleanse or quicksilver sash out of poppy's ultimate for a very specific reason:
Poppy's ultimate is NOT wholy a debuff on your champion. There are two parts:
One is a buff applied to poppy granting her certain invulnerability to other enemy units. This cannot be touched.
The other is a debuff on your champion increasing the damage you take from poppy. This CAN be QSSed.
According the video below (Riftmyths 4), no.
QSS doesn't appear to do anything. Since the video is from Cho'Gath's perspective, we can see the debuff applied and QSS does not remove it.