No they do not stack
The errata to the Dungeon Masters Guide states (emphasis mine):
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the "Combining Magical Effects" section of chapter 10 in the Player’s Handbook.
The trait talked about there states:
[...] until someone takes an action to douse the fire, the creature takes 5 (1d10) fire damage at the start of each of its turns [...]
And the Touch attack states:
[...] Until a creature takes an action to douse the fire, the target takes 5 (1d10) fire damage at the start of each of its turns [...]
These are almost identically worded and as the first does not stack with itself we can conclude that the second also does not. This can also be concluded from the fact that the features share the same name, that name being the Touch attack of a Fire Elemental. As such, while the durations of those features would overlap, only one takes effect.
Both effects would be there though
Note that the rules do not say that while the durations overlap one of the effects just vanishes, instead, it simply becomes inactive while those durations overlap. But what happens when you try to end the effect of such a condition? We actually have an entire question on this (full disclosure: I asked it):