Considering your meta as you said it is the most I can work with and I will attempt to give reasoning for everything I suggest as follows.
I will evaluate your concerns in steps of each rampant deck in your meta that you have described.
"Storm-combo"
This matchup is always dire in my opinion, you have to hope to do enough damage to make it hard for them to cast
Ad Nauseam or have the
Mindbreak Trap and hope they don't have the silence as an answer. They are simply quicker than you most times. I can see reason for your 3 main-board
Pyrostatic Pillars, but they will be dead in a lot of match-ups where you could have cast a more relevant spell. Usually a storm deck can go off turn 1 and if you don't have the
Mindbreak Trap because you kept a hand with
Pyrostatic Pillar then you would be dead in the water. Storm has many variants and I would consider which Storm is local to your meta, if it is Belcher, than I would consider
Reroute as a sideboard option, along with
Ingot Chewer as a way to stop them if they get risky and play the Belcher a turn before they can activate it. I would consider the amount of hate you put into your deck versus 1 deck type if you want to still not be losing to other archetypes.
"UW Miracles"
The match-up here is determined by what you mean by the deck name, and I would assume it is some sort of counter-top archetype, but I could be wrong. In the terms of it being what I assume it is, I would suggest some number of
Vexing Shusher because a top with counterbalance can be very hard to beat as a Burn deck. I would also consider disrupting their
Sensei's Divining Tops with
Ingot Chewer because it is uncounter-able. Running a card like
Ensnaring Bridge if their way of winning is a miracle angel variant can be very hard for them to deal with other than their usual 1 of
Detention Sphere. With
Vexing Shushers the match-up can often be much more favorable for you because they are usually a slower deck that relies on countering everything you do, without that plan, they lose the race to your combo deck.
"RUG"
I would assume this match-up is why you are running a lot of board wipers, especially uncounter-able ones at that, and why there are a lot of blasts in your sideboard. I don't mind this and I think it could be the correct way to deal with the match-up, playing around their counters and not running fetches making their
Stifles dead. I don't have much advice on this match-up and I think you are correct in your choices.
"Sneak'n'Show"
This match-up is also very hard to deal with, referencing the storm match-up. I would suggest some number of
Ensnaring Bridge or something like
Angel of Despair to deal with their
Show and Tells, along with some number of
Pithing Needle to stop their
Sneak Attack.
Pithing Needle can also be very good against a lot of miscellaneous cards in Legacy and a decent 2 of in a sideboard. I play test against a Sneak'n'Show deck myself with Burn and I find their plan against Burn is to take out
Show and Tell and put in
Through the Breach if they know you are running
Ensnaring Bridge. The match-up is not an easy one but if you can resolve a
Ensnaring Bridge it can be very hard for them to win. I am still not sure if 4 of
Sulfuric Vortex is the right number, I currently run 3 and I find that when they lay down a
Griselbrand and you have
Sulfuric Vortex they just simply can't draw cards or they will lose to a
Fireblast. I find
Sulfuric Vortex to also be very good in this match-up.
Other "modest" amounts of decks
I find Burn has a good match-up against a lot of the ones you have mentioned and you can choose whether to include other cards you notice that might be good against those decks, but usually if you can clear their creatures from the board or resolve an
Ensnaring Bridge or
Sulfuric Vortex, it can be very hard for them to win.
Ingot Chewer is also especially good against cards like
Batterskull and
Umezawa's Jitte.
Concerning suggestions given
I would only consider fetches and
Grim Lavamancer if your meta is heavy on creature based decks or tribal decks where it would be very efficient to run and also run some number of
Searing Blaze, but with the meta you described, and considering RUG being around, I like the idea of running just basic mountains and speed to keep your deck well rounded against all the quicker decks like storm and not have dead draws or lose a
Mountain to
Stifle.
Barbarian Ring can definitely win games and is good sometimes, but I find it pretty hard to run when it loses to
Stifle and
Wasteland in a deck that wants to be as efficient as possible and take advantage of that fact by running just basics and making your opponents cards dead. It can be especially contradictory to the strategy of Burn when you reach the awkward moment when you can't cast your first or second
Fireblast off it and lose a game because of it. I can definitely see reason for playing it but I like to be very precise and safe when creating a competitive deck-list.
I will leave you with a suggestive deck-list according to your concerns and good luck playing Burn!
Mainboard:
4
Goblin Guide4
Hellspark Elemental4
Sulfuric Vortex4
Chain Lightning4
Flame Rift4
Lava Spike4
Rift Bolt4
Price of Progress4
Fireblast4
Lightning Bolt20
Mountain
Sideboard:
4
Mindbreak Trap3
Vexing Shusher3
Ingot Chewer3
Ensnaring Bridge2
Pithing Needle