Qyburn

Is there any way to keep that player revived character in play?

I stole a syrio but was later informed that I couldn't put him back to shadows.

If you triggered Syrio's response, he would go back to his owners Shadows area (I think.  Better minds than mine can confirm).  It would be the same if you chose him for claim, he would return to your opponents dead pile, not yours.

As for ways to keep him in play - there's none that I can think of, because he isn't your character.  

Yup, theamazingmrg has it right. Even something sneaky like bringing in a "Cannot be Killed" character wouldn't help, because he's being "returned to a dead pile" rather than killed. 

The only tricks I can think of are making sure "that character" is no longer in play (by ensuring its not a character anymore, like, say, Venomous Manticore). Not really what you were asking, though.

Milk of the poppy on Qyburn after taking control?

Yep, NikolaP is right.  Qyburns ability creates a lasting effect that lasts until the end of the phase.  Regardless of what happens to Qyburn after that, the effect remains until it's duration has expired or the character has left play.

If it is a unique character, you could kill it and use Favored by the Warrior (AJE) to save it and search Riders of the Red Fork (FtC) to attach as a dupe, then use that dupe to save it when it would leave play.

Wait, never mind, there is no save opportunity during the end of the phase framework.