The relevant text from the SRD.
The first time it enters a creature’s space on a turn, that creature
takes 5 (1d10) fire damage and catches fire;...
So the Elemental can only "burn" a creature once per turn. It's also worthy of note that the "catching fire" ability should not stack. So if a creature is "on fire" at the start of their turn, they take a d10 fire damage, regardless of how many times they have been hit.
Also note that the elemental is a Large creature and it also has this special ability regarding moving through other creatures.
The elemental can move through a space as narrow as 1 inch wide without squeezing... In addition, the elemental can enter a hostile creature’s space and stop there.
Taken together, I read this as the Fire Elemental can freely move its 50 feet / round going through other creatures and setting them on fire. One could argue that spaces occupied by humanoids would count as difficult terrain, but given that the elemental can move through a 1 inch gap without slowing down, I would not rule that a PC is really going to "get in the way".
This attack mode of sweeping through enemies is what makes the fire elemental dangerous. But it's not a free ride. Such movement will still cause attacks of opportunity unless the Elemental took the Disengage action.
Yes, the Fire Elemental can deal a lot of damage. It's pretty scary. But all of the Elementals are scary in their own ways.