There are quite a few problems with making worn/carried items flammable. How much this impacts the game obviously depends on how often you encounter fire but assuming you encounter fire reasonably often this can make the game a lot harder. Another factor is the temperature of fire. I'm not sure if there is an official ruling on this but I presume it could probably burn cloth, silk, wood and so on but not metals.
Armour and clothes
Looking down the armour list obviously padded and hide armour would definitely burn if hit by a fireball. Leather is an odd one as in real life leather is pretty hard to burn. On the other hand it does shrink when heated and I've always assumed it would be damaged by fire. This leads to all light armours being destroyed and medium armours being at least damaged by fire spells meaning all your characters are going to have to wear heavy armour or no armour at all so anyone wanting to be sneaky or have a high dexterity is going to struggle and your giving a disadvantage to at least ranger, barbarian, bard, druid and rogue. I'm not saying it makes these classes unplayable but it can make the game unbalanced and unfair.
Even for characters who don't use armour you still hit problems. Clothes are usually flammable and nudity is often frowned upon in civilised places. Since your players are wearing mainly flammable stuff they will probably end up naked relatively often. The first one or two times unusual problems like this can be pretty fun giving the party the challenge of getting into a city and finding clothes without being arrested but if they have to do this every time they go anywhere civilised its going to get boring very quickly. This one is only a problem in campaigns where the party will meet people other than themselves. A campaign in a dungeon or far from people won't have this problem as much.
Most equipment is flammable so all that's gone if you get hit by a fire spell. So looking at the inventory of one of my low level characters he would lose a backpack, 10 torches, all his food, his longbow, and all his arrows, his rope, bedroll, the grips on his swords and various other odds and ends. All that would be left is the bare metal swords and maybe the waterskin depending on how quickly it dampened the fire. Even if most of his items survived I'm not sure how he'd carry it without his backpack.
So far casters are doing best from this rule change. They rarely bother with armour, don't need as many weapons and items and can possibly use illusions and spells to manage the nudity problem. Unfortunately any caster using a component pouch will probably lose all their components when they encounter a fire spell.
So far I have only looked at things that make the game worse for the players. But there are player advantages. Using these rules a party of wizards and sorcerers using non-flammable arcane focuses could pretty much be unstoppable by humanoid enemies. All they have to do is hit their enemies with a few fireballs and suddenly they're not facing a heavily armed army but a group of unarmed people who are currently on fire.
I have only played with this rule change once but it made the game very weird to say the least. We spent most of our gold trying to get hold of flame proof stuff and fire weapons and combat boiled down to staying away from enemies until our sorcerer fire balled them then easily wiping the floor with anyone left. I would guess that it may be possible to homebrew some rules that allowed carried objects to be flammable without ruining the game. Maybe some kind of saving throw against your worn/carried stuff getting burnt but it would be a lot of work and could end up with a very unbalanced game.