I've been trying to answer this for several years and I don't have a definitive answer but here are the options I've come up with and some comments:
Freighter travel, this is covered in the other answer and I have no good leads for who to talk to but it seems the most likely option although it'll be long.
Cruise, some cruises stop by Manila (although you may be just out of season). I can't find a cruise that hasn't already started that will stop by Manila but it's possible a travel agent might be able to find you a spot on a cruise that's already underway but hasn't yet reached here. I'd definitely go to the Star Cruises office (near Resorts World Manila, full address) and ask them, if anyone can help they can.
Private boat/yacht, you could try the Manila Yacht Club on Roxas Boulevard, I know that many private sailors do Manila to Hong Kong and they might be open to passengers if you're prepared to help. I know of no way to find out who's going where other than to turn up at the club and start asking around. Again, you might be just out of season as the rain and the storms have started and many people may not want to travel in that sort of weather.
Ferry, there are rumors that you can get a ferry or private boat from the south of the Philippines (Davao), this page has some numbers you can try but this Yahoo! answer seems to indicate that scheduled services are rare and you might have to try and find a private boat. However, while searching for that, I find news stories saying that a regular(?) service to Indonesia will start June 13 but it looks like it's just cargo. However I suspect if you got here without flying you may have come this route already. Also, for what it's worth, I should point out that the south-west of the islands are generally considered not safe for tourists -- this may be particularly true if you're having to go to small towns to get a ferry (here's the UK government advice). Having said that, I know tourists that have been and I've never heard of it being an issue.
Finally, when I asked this question a few years back on another site they did ask if it would be considered cheating to use a helicopter or hot air balloon (!). Those might be options, I think the mainland might be in helicopter range depending where you start.
I hope that helps, if you do find a way to do it please come back and answer your own question as I'd love to know how you did it.