My experience is that the MNL departure tax is collected in advance (as part of the ticketing cost) by most airlines. But NOT all. You should inquire or study the accounting that comes with the ticket.
And +1 that MNL is a dumpster fire of an airport. God save the person who has to change airlines there. You have to use the traffic packed ring road that circumnavigates the airport to go from one terminal to another. And the Philippines Airlines terminal is a sorry, sorry thing.