This last ability reads: ‘Restart the game, leaving in exile all non-Aura permanent cards exiled with Karn Liberated. Then put those cards onto the battlefield under your control.’ This means that you look at all cards that have been exiled by Karn’s abilities and separate them into permanent and non-permanent groups. Those permanents that are not auras (lands, creatures, artifacts, planeswalkers and non-aura enchantments) will end up in play. All other cards–including those auras and non-permanents that have been exiled by Karn in some way or another–will be shuffled into their owner’s library when the game restarts.
Keep in mind that other non-aura permanents that have been removed by some effect other than one of Karn’s first two abilities will not be placed in play when the game restarts. For example, a creature that was exiled by Celestial Purge will not be in play when the game restarts; it gets shuffled in with everything else.
Also, non-aura permanents that have been removed by one Karn (let’s call it Karn A) will be shuffled into a library when the ability of another Karn (Karn B) resolves. For example, lets assume that you have Karn A in play and exile some non-aura permanent cards, but this Karn A ends up leaving play. If either that Karn card or another copy of Karn enters play later, it will be a different object–Karn B–and won’t be associated with the cards exiled by Karn A. If the ultimate ability of Karn B is activated, ALL the cards that Karn A exiled will be shuffled back into their owner’s library.