"Will Obama remain in office?" One can foresee Obama refusing to leave office no matter who wins.
If Trump should win, Obama could invent "A State of National Emergency" that "local and/or state law enforcement agencies" are unable to contain, thereby allowing him to invoke martial law as provided for within the executive order he signed in September 2014. Under the provisions outlined, he as president has "the sole authority" to define what is a "State of National Emergency," how long it last and what measures may be taken to contain it. In signing the executive order, Obama created "an out" by giving him and him alone the power to declare such. Further, he can declare the emergency without any congressional interaction whatever.
One only has to look back on the actions of the last seven and half years to actually believe Obama would take such an unprecedented action. There are the multimillion dollar vacations, along with numerous other trips under whatever guise made at taxpayer expense, Secret Service protection 24/7, Air Force One, 24/7 "room service" and the other trappings of office. He is not about to give all this up if he has the ability within his grasp not to.
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Perhaps this may sound unbelievable, yet one only has to look back at all of his actions and remember he "has a pen and telephone," not to mention his disdain for the Constitution and the Congress as well as the Judiciary. Obama has throughout his tenure seen the other two branches of government as mere nuisances to his agenda.
Should Obama elect to invoke the powers contained within the executive order, he will in turn create a constitutional crisis unmatched in the history of the republic.