The Philippine government allows citizens of 145 countries to enter the country visa-free for visits of 21 days or less, and citizens of a few other countries for visits of 7 days or less; 59 days or less. The list of these countries can be found in the Philippine government portal,
http://www.gov.ph/faqs/visa.asp . For visits longer than 3 weeks, a Philippine visa will be required though.
A few countries allow Filipinos to enter their country visa-free too for short visits. Among these are Singapore, Thailand, Malaysia, Indonesia, Vietnam, Laos, Cambodia, Brunei, East Timor (as fellow members of ASEAN); Hong Kong, Macau, Seychelles, Maldives, Nepal; Cuba, Netherlands Antilles, Bermuda, Haiti; Costa Rica, Nicaragua, Panama, Ecuador, Brazil, Peru, Colombia, Bolivia, El Salvador, Honduras, Suriname, Belize; Morocco, Israel; Palau, Fiji.
(Thanks to Robby G. for this list!)
One proposal made by some people is "reciprocity" in visa-free entry; ie, if the Philippine government gives visa-free entry for short visits to nationals of about 150 countries, then governments of those 150 countries should also give Filipinos visa-free entry into their lands. I don't believe in reciprocity. Just open entry of foreigners to the Philippines visa-free (again, for visits of 3 or 4 weeks or less), even if the governments of those foreigners will not grant the same privilege to Filipinos. I advocate full free trade, even unilateral trade liberalization, so I also advocate full mobility of people, even unilateral free entry and mobility of foreigners into the country -- except the known terrorists and organized criminals.
The geographical isolation of the Philippine archipelago from the rest of the world, even from its South East Asian neighbors, is already a big factor why the country is not attracting too many tourists. For instance, when you land in Vietnam, you can take a bus or train to China and Laos (north), bus or train to Cambodia (west), then further west to Thailand, Myanmar, etc. But when you land in Manila or other international airports (Cebu, Davao, Clark), we have no other neighboring countries accessible by land. It's the vast Pacific Ocean in the east, South China Sea in the west, and lots of small seas in between the islands and provinces. So, easier access through visa-free entry is one incentive to lure tourists, foreign businessmen and visitors, to come in. Although it doesn't help that Manila's international airport (NAIA 1 and 2) are just "provincial and domestic" airport in size and facilities compared to many other countries' international airports, like those in Thailand, Singapore, Malaysia, Vietnam, Korea, Japan, and so on.
Some African countries which don't give visa-free entry for Filipinos are ruled by dictators who just live off on taxpayers' blood and sweat, both local and foreign taxpayers. Even if the "visa fee" is only $10 or $20, for those dictators and bureaucrats, that's still money that can go to their pockets. Now if their rich nationals (say, families and relatives of those dictators, top bureaucrats and private cronies) would come into the Philippines to see Boracay, Davao, Cebu, Manila, Baguio, Bohol, Palawan, Puerto Galera, etc., let them come in visa-free, let them spend their money in Philippine beach and mountain resorts, in Philippine malls and restaurants, in Filipino boatmen and travel guides. (We can't bar and confiscate their money as "ill-gotten wealth" as it is an internal problem that has to be resolved in their own countries.)
The wealthy countries of W. and N. Europe, N. America, N. Asia (Japan and Korea) are too protective of their welfare system; that system is too expensive, financed by high and multiple taxes of their citizens, and continuously accummulating public debts (e.g., France's unfunded social security system is around 200% of GDP!) that they can't extend such welfare to many citizens of other countries. Hence, the strict "visa-required" rule, even for few-days visits.
The Department of Foreign Affairs (DFA) though, does not give visa-free entry yet to citizens of Russia, Kazakhstan, Croation, other central Asian and eastern European countries. I don't understand why, but it will be an important diplomatic and economic move to lift such travel restriction and allow the nationals of those countries to come in visa-free.
People by nature want to be free to move around in many places, islands and continents. Before, it was the harshness of nature and environment (very hot desert, very cold and prolonged snow, presence of many savage animals that can attack people, tribes that are averse to entry of foreigners) that prevented many people to move around. Later on, governments of many countries began expanding, and so the bureaucracy and restrictions to travel also expanded. With modern technology and increasing competition among international airlines, international hotels, international telecomms companies, communication and mobility among friends, families and clans, business and cultural partners, become easier. It is only rightful that full mobility be respected and given back to people -- except to those who inculcate hatred and violence.