I am a US permanent resident but I am a Filipino citizen. I am planning to travel to the Philippines and wanted to get a ticket with a 15 hours layover in the UK. Am I allowed to go out of the airport and have a tour to London?
You can transit without visa (TWOV) landside (i.e., you are allowed to leave the airport). Citizens of Philippines can TWOV.
It is a good idea, however, to have some supporting documents (e.g. work contract) available, be dressed reasonably well etc., because TWOV is a concession.
The quick answer is no without a visa. As a Philippines national you need a visa to go through the uk border.
This is clearly explained by the uk border agency here http://www.ukba.homeoffice.gov.uk/visas-immigration/transitthroughtheuk/visa-to-transit-landside/
Check the "Are you a visa national" section under "more information"