Will a U.S. citizen living and working in the Philippines have to pay U.S. tax on contributions and gains for investments through Pru-Life?

If it is in a US variable-life insurance policy or an offshore unit-linked life or insurance-wrapped portfolio bond, you would not pay until you removed the money form the policy.  If it is offshore, you many need to file a FATCA form if you haven't already.  

US citizens are taxed by the US on their worldwide income. While living in the Philippines, you'll report any such taxable transactions on your US return the same as if you were living in the US. You may already know that the Foreign Earned Income Exclusion (FEIE) applies only to earned income and won't help you with "passive" income if it's otherwise taxable by the US.
So then, for example, if your Pru-Life account is a traditional IRA, withdrawals are taxable. If it's a life insurance policy paid for with after-tax dollars, then withdrawals of cash value are tax-free up to your basis in the policy. If you're not receiving any taxable distributions and wonder what happens with funds that remain in the account/policy, then there'd typically be no tax reporting,
Cross-border taxation is complex and there may be nuances to your situation that aren't covered in your question. Treat this as general information and consult a cross-border tax specialist if you need specific individualized advice beyond the general thoughts I've shared here. I hope this helps.