Elemental gold is not flammable. Flammability requires elements to be able to unite with oxygen in a combustion reaction. The inert electronic configuration of gold makes it unreactive with oxygen even in molten form.
The presence of gold in its native, atomic state in nature indicates the high inertness of this element. Even billions of years spent in oxidative and reductive atmospheric conditions in different geographical epochs did not cause this metal to react and form compounds.
The special inertness of gold results from its having a single outer electron in an s orbital. Because the s orbital can accommodate a maximum of two electrons, this outer shell is half full. Half-filled outer shells are especially stable, because this outermost electron is not affected by the quantum and columbic forces that electrons in partially filled shells exert on one another. This stability makes unlikely gold's participation in any chemical reactions, justifying its presence in native form.
More electronegative elements than oxygen, such as the halides, have higher electron affinities and are able to attract this outermost electron from the s orbital of gold, forcing it to react. Once gold loses this stable, half-filled structure, it becomes able to lose even more electrons, which is why oxidative states of Au(I), Au(III) and Au(V) are all possible.