Q:

What damage does one’s clothes bursting into flame deal, and what is needed to put it out? 

The Phoenix Soul Sorcerer can ignite flammable objects like cloth with a touch.   How should I handle this when a player playing such a character in one of my games ignites an enemies clothing?   Specifically, what damage should being on fire deal and what should be necessary to end the effect?   I'm looking for an answer that works well with existing material; it's not important that the answer be any sort of published rule.  
As part of an answer, please indicate what you are balancing this use of the ability around and why in your adjustments to its function, so that I can understand where the suggestion is coming from when using that sort of approach to adjudicate the class ability. 

A:

There are no published rules, but...
You could use some of the official rules in the  Adventuring Gear section of the Player's Handbook

[Oil]
... You can also pour a flask of oil on the ground to cover a 5-foot-square area, provided that the surface is level.  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. 
[Alchemist's Fire]
...A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames. 

I would probably modify this somewhat to scale up to 5 damage with level since it is a class feature without limited uses, so I would have it deal Charisma modifier for 2 rounds unless someone puts it out. ( Perhaps more rounds on a Large or larger creature). 
This makes it somewhat comparable to a monk's Martial Arts (unarmed strike) damage per attack.  Estimates by tier: 5.5, 7.5, 9.5, 10.5 for monks vs 6, 8, 10, 10 for this variant rule. 
Disclaimer
Unearthed Arcana is playtest material and not necessarily balanced for the core system.  Most effects that ignite objects don't affect those worn or carried by other creatures.  See fire bolt (emphasis mine)

A flammable object hit by this spell ignites if it isn't being worn or carried. 

It is likely that worn objects would not be eligible if this archetype were officially released.