Clarifying littleadv's answer a bit, where he says, "As a US citizen - you're always considered a US resident for tax purposes."
This is not quite true. What I think he is trying to say is that you are always under the jurisdiction of the US tax code and subject to US tax law. Tax law does, however, differentiate between citizenship and residency, and residency considerations are quite important if you are abroad.
In general, the test of foreign residency (along with big tax deductions) is whether you are outside the US for virtually all the time (the physical presence test, iirc 330 days a year in foreign countries) or the residency test (where you are legally resident and whether you have made representations to any government anywhere that you are a US resident). If you pass the tests for foreign residency on either test, you may be eligible for income tax deductions but not FUMA/FICA tax deductions.
Since these tests involve residency, including legal residency, the only way your billing address in the US could become an issue is if you represent yourself as residing there to any government anywhere in the world. If it is always only a billing address, and never presented as an address of residency that should be good, but if you ever say you reside there then you have to pass the physical presence test to be a foreign resident.
So the answer is "no." However it can result in trouble if you aren't careful about how you present yourself.
For example, if you come back to the US for two months or so and tell the state government (maybe to renew your state driver license) that you live at your billing address, that could jeopardize your foreign tax exclusion. The mere presence of a US billing address has no effect, but how you present it in many cases might.