Based on what I know of Obeka, Brute Chronologist, I anticipate that Warkite Marauder's affect will be permanent when used with Obeka. But, it seems so good, I just had to double check. A "pirate-y" way to deal with Eldrazi, if you know what I mean...
No, the effect ends during the cleanup step, same as always.
717. Ending Turns and Phases
717.1. Some cards end the turn. When an effect ends the turn, follow these steps in order, as they differ from the normal process for resolving spells and abilities (see rule 608, “Resolving Spells and Abilities”).
717.1d The current phase and/or step ends. If this happens during combat, remove all creatures and planeswalkers from combat. The game skips straight to the cleanup step; skip any phases or steps between this phase or step and the cleanup step. If an effect ends the turn during the cleanup step, a new cleanup step begins.
514. Cleanup Step
514.1. First, if the active player’s hand contains more cards than their maximum hand size (normally seven), they discard enough cards to reduce their hand size to that number. This turn-based action doesn’t use the stack.
514.2. Second, the following actions happen simultaneously: all damage marked on permanents (including phased-out permanents) is removed and all “until end of turn” and “this turn” effects end. This turn-based action doesn’t use the stack.
Private Mod Note
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Why bother with mere rulings when so many answers can be found in the Rules?
No, there is no way to extend Warkite Marauder's effect after the current turn using just Warkite Marauder and Obeka.
If you remove an artifact's abilities and change its power and toughness with Warkite Marauder, then end the turn with Obeka, this will move the turn to the cleanup step (C.R. 717.1d; see also C.R. 717.2e), and the effect of Warkite Marauder's triggered ability will end during that cleanup step (C.R. 514.2). It doesn't matter when the turn ends this way; even if you somehow manage to end the turn with Obeka during a cleanup step, this will only move the turn to another cleanup step and the effect will end during that new step (C.R. 717.1d, 514.2; see also C.R. 514.3a). This applies in general to any effect that lasts "until end of turn" or "this turn"; see also this thread, this thread, and this thread. Compare this with an ability that says to do something "at the beginning of the next end step" or "at end of combat"; see also this thread and this thread.
If you remove an artifact's abilities and change its power and toughness with Warkite Marauder, then end the turn with Obeka, this will move the turn to the cleanup step (C.R. 717.1d; see also C.R. 717.2e), and the effect of Warkite Marauder's triggered ability will end during that cleanup step (C.R. 514.2). It doesn't matter when the turn ends this way; even if you somehow manage to end the turn with Obeka during a cleanup step, this will only move the turn to another cleanup step and the effect will end during that new step (C.R. 717.1d, 514.2; see also C.R. 514.3a). This applies in general to any effect that lasts "until end of turn" or "this turn"; see also this thread, this thread, and this thread. Compare this with an ability that says to do something "at the beginning of the next end step" or "at end of combat"; see also this thread and this thread.