I don't know about state income tax, but someone mentioned that US citizens have to pay federal income taxes regardless where they live. In general, they have to report their income (and in most cases, also their accounts in non-US financial institutions - aka FBAR).
However, whether they actually have any tax liability depends on other factors, such as the Foreign Tax Credit, which someone else mentioned.
There is also a significant Foreign Earned Income Exclusion (up to $99,200 for 2014), although it is not relevant if you are retired and your income is passive.