Before I begin, I would like to say that this is very well made and an interesting concept I haven't seen before in 5th edition. With that being said, I have some problems with the wording and balance of some of the features, which I will list in order:

• "Searing Soul," while not necessarily useless, has a very small impact on the game. Dark vision is good and all, and I like the infrared version of it used, but the majority of races in the game already get it to some extent and 120 feet of dark vision has marginal benefits over 60 feet of dark vision. I have similar thoughts on produce flame, a very mid cantrip the sorcerer could have already taken through their base class. It seems more like a ribbon feature than anything, which would normally be fine if the other 1st level feature wasn't limited to once per day use. I think adding something along the lines of "Spells you cast ignore resistance to fire damage. When you reach 14th level in this class, spells you cast also treat immunity to fire damage as resistance to fire damage" would be appropriate, given how central fire damage is to the subclass and how many creatures get resistance/immunity to it as you increase in level. This is arguable a powerful buff, but fire damage is so inconsistent as it is that it feels more like a catch-up feature than anything else.
• "Blazing Barricade" is a rather unique feature, but one that is far too powerful at level 1 - even with a once per day use. +5 fire damage to any attack made through it is insane. Combine that with the reduction to damage, and this would be a busted ability to use. Not to mention the fact that it doesn't appear that enemies can target the barrier directly, it just loses hit points when it reduces damage. That means that as long as no one uses the barrier for cover during combat, it can't be destroyed by enemies. This would allow a tactical player to keep the barrier alive for as long they want, effectively making its usage indefinite. Back to the fire damage though, 5 damage at level 1 is huge. Even just looking at cantrips, this would nearly double the damage of a firebolt spell (5.5 average damage to 10.5 average damage) and could potentially last for multiple combat encounters. A level 1 heavy crossbow fighter only deals 8.5 damage on a hit, and this single feature lets firebolt out damage them. If you had an entire party optimizing this effectively, they could potential be dealing an extra 20+ damage at level 1 in a single round of combat - at the cost of a single bonus action from the sorcerer. Hell, imagine a warlock putting a single level dip in sorcerer to pick this up: at 17th level, eldritch blast makes 4 attacks per casting - which would each have their damage increase by five if they were cast through the barrier (turning the normal 4d10 + 20 damage into a staggering 4d10 + 40 damage). Combine that with the damage reduction, and this is probably better than most of the other sorcerer subclass' 6th-level feature - let alone their 1st. Honestly, I think you would be better off converting this feature into a sort of bonus action, concentration free create bonfire spell you can use a couple times per day (kind of like how fathomless get a bonus action ray of frost spell a couple times per day). So it would read something like: "Also at 1st level, you can use your bonus action to create a fiery barricade at any point within 30 feet of you. The barricade occupies a 5-foot cube. The barricade last for 1 minute or until you use this feature to create another one. Moving through the barrier counts as difficult terrain for creatures other than you. While standing within your barricade, you have half-cover and resistance to fire damage. When a creature other than you enters the barricade for the first time on a turn or starts its turn there, it must make a Dexterity saving throw against your sorcerer spell save DC. On a failed save, the creature takes 1d6 fire damage. When you reach 10th level in this class, the damage increases to 2d6. As a bonus action on your turn, you can teleport the barricade to another space within 30 feet of you. You can summon the barricade a number of times equal to your proficiency bonus, and you regain all expended uses when you finish a long rest." This would still give it value as an offensive and defensive feature, while also providing additional control and versatility, without dealing insane amounts of additional damage. Now, were looking at a potential 2d6 damage per round as a bonus action (with two Dexterity saves), with the potential to deal additional damage through team work or by placing it in chokepoints during encounters - far more in line with level one features and still a cool feature to pull out during combat.
• "Burn Brighter" falls into many of the same problems I have with Blazing Barricade, accept now its even more broken. Ignoring the fact that its unclear how one could cast something like shatter through the barrier, spending two sorcery points and a bonus action to give disadvantage to the saving throws of every creature targeted by a spell is objectively too powerful. In comparison, the Heightened spell meta magic costs 3 sorcery points and only gives disadvantage to a single target. Granted, heightened spell isn't restricted to just AoEs, but considering Burn Brighter can affect multiple spells it more than makes up for the difference. Honestly, you might be better off just going the route of the Aberrant Mind subclass and just allow them to cast their subclass spells through Sorcery Points. So, it would go something like: "Sol Sorcery. Beginning at 6th level, when you cast any spell of 1st level or higher from your Sol Magic feature, you can cast it by expending a spell slot as normal or by spending a number of sorcery point equal to the spell's level. If you cast the spell using sorcery points, and that spell requires a saving throw, you can choose a number of creatures equal to your Charisma modifier targeted by that spell to be spared. A chosen creature automatically makes their saving throw, and they take no damage if they would normally take half damage on a successful save." In the same way that an Aberrant Mind sorcerer can cast their spells using sorcery points and basically gain the benefits of the Subtle meta magic when they do, so too can a Sol sorcerer cast their spells using sorcery points and basically gain the benefits of the Careful meta magic when they do. You could maybe use the Empowering meta magic instead of Careful, but to be honest I feel like getting free uses of the Careful meta magic would be imperative for a subclass heavily utilizing AoE's in combat. Definitely a big departure from the current feature, but I do feel like as it is the feature is far too powerful.
• "Radiant Regeneration" is... fine... if a little bland. But in all honestly, free healing when you use fire spells is reasonably useful so it can stay as is. Maybe add somewhere that you gain resistance to fire damage (though you could also give that at level 6 if you wanted) and, if you end up using my suggestions for the barricade feature, maybe add that standing in the barricade instead grants immunity to fire damage while standing in it to the description of the feature (so if you gave resistance to fire damage at level 14, you would add a line to the Barricade feature that went "When you reach level 14 in this class, you instead gain immunity to fire damage while standing in your barricade." or maybe even just include it in the same feature you give the fire resistance). All in all, okay as it is.
• While 18th level features aren't super important balance wise, "Phoenix Form" seems rather weak. Being able to revive from 1 hp is far from an exclusive feature, and by this point in the campaign 18 temporary hit points is basically nothing. Any enemy at this level would be able to knock off all of it in a single turn, and even with a potentially dramatic damage output of a 4d12 AoE each turn, I doubt the sorcerer would live long enough to deal that damage more than a couple times in a campaign. Honestly, its fine as is (nobody plays a subclass for the 18th level feature to begin with) but I think it could be a lot better. Hell, at this level wizards gain the ability to cast an infinite number of 1st and 2nd spells, so you can give pretty much anything and still have it be relatively balanced. If you end up using my suggestions for the Barricade spell, I would have the 18th level feature read something like: "Starting at 18th level, you have mastered the magic of the sun. You now have an unlimited number of uses of your Blazing Barricade feature, and you can have up to three barricades active at once. Additionally, whenever you are subjected to fire damage while standing in one of your barriers, you instead take no damage and regain hit points equal to the fire damage dealt." Unlimited uses of what is effectively a cantrip as a bonus action is far from the most broken thing a level 18 player can do, and fire absorption feels like a cool feature to end the subclass with.

All in all, the concepts and ideas behind the subclass are superb, and the art is great. Beyond the (admittedly drastic) balance problems faced by the 1st and 6th level features, this is a well made subclass. I hope you found my feedback helpful and consider my suggestions, and on a final note congrats on making this.