Yes, you have to pay US income tax on money that you earn in Belize. However, there are exclusions that you can claim such as the Foreign Earned Income Exclusion, which will allow you to exclude a certain amount of money (around US $100,000 each) for you and about the same amount for your spouse to exclude that for tax in the United States. However, if your earned income is above that, then you have to pay income taxes in the United States.
If you are paying BZ taxes on income in Belize, like I do in my rental income, I pay taxes here but when I do my income taxes for the States, I get to deduct the total amount of my income tax here off the total amount of income tax I owe in the States. You would always want to consult a US tax attorney before making any decisions about income taxes in the US and in Belize. The best attorney here in Belize is Wrobel & Co. (Ryan Wrobel is a contributor to Best Places and can be found on this site.)
(Wrobel & Co. Attorneys at Law, Belize, pictured.)