Oath of Druids triggers. It is not activated. In this case, the card is left on top of the library. This is because Grafdigger's cage prevents it from entering the battlefield at all so Containment Priest's replacement effect doesn't have anything to replace. It stays on top because Oath of Druids' ability says to put all "other" cards in the graveyard. Even though the creature can't enter the battlefield, it is still not part of the "other" subset of cards so it won't go anywhere.
Dredge and Spirit of the Labyrinth is different because of what is being replaced and it depends on whether a card has been dredged or drawn. For example:
If a card with Dredge is in a player's graveyard and they have not drawn a card yet, they can Dredge that card and get it back into their hand. Since they replaced their draw with dredging, they have not actually drawn a card yet. So, they can then dredge another card (if they have one and they were to draw a card) and continue to do so until they actually draw a card for turn.
Once that player draws a card however, they can no longer dredge that turn. This is because Spirit of the Labyrinth prevents additional draws. Since Spirit prevents the draw entirely, there is no event to replace with Dredge.
In general, if something is impossible, it can't be replaced (C.R. 614.7, 101.2, 101.3; see also C.R. 614.17c). If Grafdigger's Cage keeps a card from entering the battlefield, Containment Priest can't replace it with being exiled. Thus, in the case of Oath of Druids, the last revealed card stays on top of the library (rather than somewhere else, since the card is only revealed [C.R. 701.14b]) and "all other cards revealed this way" go to the graveyard. And if Spirit of the Labyrinth hinders drawing, it can't be replaced with dredge (or any other effect that would replace drawing a card).
EDIT (Sep. 30): One rule was renumbered with Kaladesh.
EDIT (May 4): Added citation to a rule added in Amonkhet.
EDIT (Aug. 5, 2017): Added rule citation.
EDIT (Sep. 28, 2017): One rule was renumbered with Ixalan.
The replacement effect changes the event before it would be prevented?
In order to be prevented from entering, it has to try to enter. It goes to exile instead of trying to enter, so cage doesn't have an effect.
Why is it such and how does it compare to Dredge cards and Spirit of the Labyrinth?
Dredge and Spirit of the Labyrinth is different because of what is being replaced and it depends on whether a card has been dredged or drawn. For example:
If a card with Dredge is in a player's graveyard and they have not drawn a card yet, they can Dredge that card and get it back into their hand. Since they replaced their draw with dredging, they have not actually drawn a card yet. So, they can then dredge another card (if they have one and they were to draw a card) and continue to do so until they actually draw a card for turn.
Once that player draws a card however, they can no longer dredge that turn. This is because Spirit of the Labyrinth prevents additional draws. Since Spirit prevents the draw entirely, there is no event to replace with Dredge.
EDIT (Sep. 30): One rule was renumbered with Kaladesh.
EDIT (May 4): Added citation to a rule added in Amonkhet.
EDIT (Aug. 5, 2017): Added rule citation.
EDIT (Sep. 28, 2017): One rule was renumbered with Ixalan.
It appears the Oath creature got exiled.
The replacement effect changes the event before it would be prevented?
In order to be prevented from entering, it has to try to enter. It goes to exile instead of trying to enter, so cage doesn't have an effect.
Or it's an online bug?
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