The Monster Manual (page 125) says for the Fire Elemental under Touch:
[...] If the target is a creature or a flammable object, it ignites. [...]
The Stone Golem on page 170 is a creature and does not have immunity or resistance to fire. However, stone isn't very flammable.
Does the Stone Golem ignite if hit by a Fire Elemental?
Context: We had a short discussion during our last game session. GM ruled that the Golem ignites but a player disagreed, because (most) stone doesn't burn.
First, there's the rules perspective. As you pointed out, the Fire Elemental ignites creatures and flammable objects. The Stone Golem is a creature. Therefore it ignites.
Second there's the in-fiction perspective, which is what the player was challenging. The Stone Golem is a lump of rock(s) animated by magic, but the Fire Elemental is literally the essence of fire itself. If it wants to burn something, then by golly that thing is going to burn. The Fire Elemental is essentially putting a sort of clinging magical fire onto the golem.
Alternately it may be the case that it's the magic that animates the golem that's being burned. As vaxquis pointed out in a comment, even creatures that are completely immune to fire are ignited by the Fire Elemental's touch, they just don't take any damage from it. The fact that the Stone Golem doesn't even have resistance to fire, much less immunity, suggests that something about it burns just fine.
[...] If the target is a creature or a flammable object, it ignites. [...]
A Golem is a creature (specifically a construct).
So it will ignite. The flammable question only becomes relevant if you are talking about objects.
