As you asked about DUAL CITIZENSHIP and not Dual Residency, you should know that the US and the Philippines have a tax treaty in place.
Generally citizens are taxed on all income from all sources. Under the treaty (Relief from DOUBLE TAXATION) each country will grant a credit to the citizen based on the actual residency.
In accordance with the provisions and subject to the limitations of the law of the Philippines , the Philippines shall allow to a citizen or resident of the Philippines as a credit against the Philippine tax the appropriate amount of taxes paid or accrued to the United States. Such appropriate amount shall be based upon the amount of tax paid or accrued to the United States, but the credit shall not exceed the limitations (for the purpose of limiting the credit to the Philippine tax on income from sources within the United States, and on income from sources outside the Philippines) provided by Philippine law for the taxable year.
For the purpose of applying the Philippine credit in relation to taxes paid or accrued to the United States, the rules set forth in Article 4 (Source of Income) shall be applied to determine the source of income.
For purposes of applying the Philippine credit in relation to taxes paid or accrued to the United States, the taxes referred to in paragraphs (l)(a) and (2) of Article 1 (Taxes Covered) shall be considered to be income taxes.
The treaty (to simplify) says that the country you actually are living in will let you claim a credit for taxes paid in that country that is also on income taxed in the other country. Each country has agreed to do this.
As a citizen of each you will need to report your income but you will not be taxed in both without relief under the treaty by use of the credit.
Please let me know if you need more information about the tax treaty.