Say a character was in a burning house, the fire not caused by magical means, and had to pass through a hex of fire. What is the fire damage for this in 5e? What about per round?
There isn't an explicit rule for this -- it is left to the DM to improvise an amount of damage that seems appropriate for the situation.
The DMG (p. 249) gives some guidelines on improvising damage. It suggests that falling into a fire pit might cause 2d10 damage.
In the case of someone running through a small area of fire as per your scenario, I'd look to the create bonfire cantrip from the Elemental Evil supplement:
You create a bonfire on ground that you can see within range. Until the spells ends, the bonfire fills a 5-foot cube. Any creature in the bonfire’s space when you cast the spell must succeed on a Dexterity saving throw or take 1d8 fire damage. A creature must also make the saving throw when it enters the bonfire’s space for the first time on a turn or ends its turn there.
Even though that is magical and not mundane fire, this suggests that a dextrous character could pass through a 5-foot cube of fire without taking damage, or with fairly minimal (1d8) damage if they're clumsy. Think about passing your finger through a candle flame without getting burned. I think I'd probably improvise pretty close to this for small areas of mundane flame. For a larger area, I'd probably do something like 1d10 per round in the flame.
I think the rules for Oil in the PHB (pp. 152) might be what you're looking for as it is 1) not caused by magical means and 2) gives the damage for passing through a hex of fire.
If lit, the oil burns for 2 rounds and deals 5 fire damage to any creature that enters the area or ends its turn in the area. A creature can take this damage only once per turn.
Another point of reference: PHB p. 288, Web spell. "Any 5-foot cube of webs exposed to fire burns away in 1 rouund, dealing 2d4 fire damage to any creature that starts its turn in the fire." (emphasis added)
Again, it's roughly 5 HP damage per round to be in a space that's on fire.
Per DQV rules (A healthy mix of reality and RPG)
Useful Temps to know
212 F Boiling Water D4 +2 Per Pulse
300 F Paper Ignites
450 F Dry Wood Combustion 570 F Boiling Oil D6 +4 Per Pulse (Pint of oil works here but lasts shorter) 450 F Tin/ Lead Melts 1,500 F Middle of Typical Fire D8 +4 per Pulse 2,000 F Silver & Gold Melts 2,500 F Lava D10 +20 Per Pulse 3,000 F Iron / Steel Melts
Next Four are Magical
3,000 F Middle of Magic Fire
3,500 F Valerian Steel Melts
4,000 F Estimated Dragon Breath 5,000 F Saurons Jewelry 2d10 +20 10,000 F The Sun 4d20 +20 50,000 F Lightning 20d20 +20