I remember back in 1986 how they were calling the Red Sox-Mets World Series "the Yankee fan's worst nightmare." The team's AL East rival taking on the team they share the same city with. I honestly believed going in that the Yankee fans hated the Red Sox more, so you'd be seeing most of them (probably silently) pulling for the Mets.
I could not have been more wrong. Among a core group of Yankee fans I knew back then, I was shocked to see many openly hoping to see the Red Sox take it. (All that "curse" BS had not infected the landscape yet.) The hatred for the Mets was definitely there, and pretty blatant, deep and vocal.
I came to the conclusion that while the Red Sox were the hated rival, the team who occupies the same city and who Yankee fans interact with more was definitely more despised. I remember seeing George Steinbrenner interviewed right after that World Series and congratulating the Mets on their win. (And I knew that had to be killing him to be so magnanimous, as I also recall how much he wanted to beat the Mets so badly in spring training games back in the 1970s.)
The Mets and the Phillies have been rivals, but not really for very long. It only really got going in 2007, when the Phillies staged that incredible comeback to take the NL East on the final day. For years, either the Mets were good and the Phillies were bad (like in 1969 and the early 1970s, and the late 1980s and late 1990s), the Phillies were good and the Mets were bad (the late 1970s, or the early 1990s) or both teams were bad. So there never really was a time when both teams went at each other in a pennant race to really form a heated rivalry before 2007.
So far what I've heard from Mets fans has been across the board. Many are treating it like the Iran-Iraq War of the 1980s ("I hope they end up killing each other"). But I would still bet that most will end up hoping (many silently) that the Phillies end up winning it all. (And besides, wouldn't you want to see the former Met pitcher pictured here win another World Series ring at the Yankees expense? I sure would.)
I would bet many will be going along the lines of what my father once said years ago, and still lives by even today:
"Anybody but the Yankees."
Phillies in six.