Falcar
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The weapon would become a +2 Flaming weapon. Greater magic weapon only adds an enhancement bonus to the weapon. It does not change it. if the bonus the spell would give is +2 then the normal +1 is overlapped and the higher one is used. A +3 weapon would have nor benefit from GMW. That said GMW is helpful to the person with a +1 flaming burst holy Animal bane blade because it makes the enhancement bonus higher (+1-+5).
Coinshot Colton
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Related question, what happens if you have a +1 Furious weapon instead?
So, +1 Furious weapon is given GMW (let's say the GMW is at CL 8, so it gives +2), and the person wielding it is raging. In this case, is the weapon:
A) The weapon is +3 because Furious only adds +2 to the original +1
B) The weapon is +4 because Furious is added onto GMW