Jealous Eyes reads "When this card is played, immediately end and resolve the current action chain. Action: Move up to 2 resource tokens from target opponent's card to a card you control."

My question is, is the action part of Jealous Eyes considered to be part of the 'current action chain' or does it constitute the start of a new action chain? For example, I play it targeting my opponent's Slave Pen. Can he respond to my action by playing Burn It Down, thus denying me the two resource tokens?

No he can't beause the chaion is over and gets resolved, so that you get the token. After that your opponent can open a new chain and destroy his slave pen, if he still wants to

Hehe, thought as much, thanks.