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If I exile cards with a collection counter on using Evelyn, the Covetous's ability, can I still play those cards if Evelyn is not on the battlefield?
The wording on cards such as Gonti, Lord of Luxury and Thief of Sanity read: "You may look at and cast that card for as long as it remains exiled..."
The wording on Evelyn, the Covetous suggests that she has to be on the battlefield.
You are correct that Evelyn's ability is handled differently than Gonti's.
In both cases, a continuous effect is generated that allows you to cast the exiled card(s). These effects fall under the purview of comp rule 611.
In the case of Gonti and the like, the continuous effect is created by the resolution of a triggered ability, and so falls under rule 611.2:
611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”).
For Gonti, the stated duration is "as long as [the card] remains exiled". Until the card leaves exile, the continuous effect will persist even if its creator is no longer on the battlefield.
Evelyn, in contrast, has a static ability that creates a continuous effect, which is handled by rule 611.3:
611.3. A continuous effect may be generated by the static ability of an object.
611.3b The effect applies at all times that the permanent generating it is on the battlefield or the object generating it is in the appropriate zone.
This static ability requires Evelyn to be on the battlefield to function. As soon as she leaves, the static ability and its related continuous effect go away.
