Can USA State sales tax be avoided for products purchased in the USA and exported to Belize

If you purchase the phone in a state that has no sales tax, yes. Be aware a phone purchased here will have no warranty support outside the U.S. If anything happens to it that is a warranty covered issue, it would have to be shipped back to the U.S. to have it fixed.

States with no statewide sales tax are:

Delaware

Montana

New Hampshire

Oregon

Alaska

They all do have have sales tax on some items in certain locations.

Otherwise, buy the phone online from any vendor that doesn't charge sales tax and wait for it to be shipped to you. Legally, that doesn't get you out of paying the state tax. You are obligated to pay Use Tax on items (you paying the uncollected sales tax) that would be taxed had you purchased it locally.