Most things you would encounter on ancient battlefields don't burn. People and metal armour are difficult to set ablaze; there are far easier and more straightforward ways to destroy them. Fire was therefore a much less common feature of ancient land warfare than movies would have you believe. There are really only two situations in which the ability to project fire (whether by arrow, artillery bolt/stone, or with a flamethrower) would have potential to affect the outcome of a fight.

The first is a siege assault in which the defenders' fortified position is built of wood. Makeshift forts and the walls of smaller settlements would often take the form of a simple palisade, and in those cases fire could be used to level the defences. For this purpose, it seems the Greeks invented the flamethrower about a thousand years before they invented Greek fire. A basic flamethrower was used by the Thebans to reduce the Athenian base at Delion in 424 BC:

They sawed in two and scooped out a great beam from end to end, and fitting it nicely together again like a pipe, hung by chains a cauldron at one extremity, with which communicated an iron tube projecting from the beam, which was itself in great part plated with iron. This they brought up from a distance upon carts to the part of the wall principally composed of vines and timber, and when it was near, inserted huge bellows into their end of the beam and blew with them. The blast passing closely confined into the cauldron, which was filled with lighted coals, sulphur and pitch, made a great blaze, and set fire to the wall, which soon became untenable for its defenders, who left it and fled; and in this way the fort was taken.

-- Thucydides 4.100.2-4

The same device was used by the Spartan general Brasidas around the same time, to force entry into the Athenian-held town of Torone. But this weapon was a gimmick and most besieging Greek armies would resort to more traditional ways to set things on fire (i.e. torches). After the two instances in Thucydides' account of the Peloponnesian War, the flamethrower is not heard from again.

The second, much more prominent, much more common, and much more obvious situation in which fire could be useful is naval warfare. All ancient warships were made of wood and their seams were probably sealed with bitumen. Ships were kept in drydock because waterlogged ships were slower and less manoeuvreable. In other words, ancient fleets were extremely flammable. Fire was a great threat to them. The ability to throw fire at them from a distance gave Byzantine fleets a huge advantage.

In short, since flammable objects (ships) were much more common in naval warfare, it made sense for Greek fire to be primarily used by the fleet. On land it is difficult to imagine a use for it that couldn't be as easily achieved by other forms of violence, which also had much less potential to get out of hand and harm both sides indiscriminately.