Burn Mage is a Freeze Mage variant without Alexstrasza, which uses spell-damage boosted spells to "burn" your opponent down. Much like Freeze Mage, Burn Mage is an intricate combo deck that is difficult to play optimally and may take some time to get used to. That said, the precise combo turns and unlikely comebacks are what make the deck an exciting one to try out in the current meta.
Credit where credit is due, this deck is an updated and "standard"-ized version of Jabba's OTK Mage, sharing 26 cards in common with that decklist. The substitutions made were Kobold Geomancers to Cult Sorcerers, as well as second copies of Doomsayer and Blizzard added for deck consistency.
Mulligan is simple, Forgotten Torch and card draw; however, do not keep Coldlight Oracle unless you are playing against a Warlock deck which aims to Life Tap for the first few turns. If you have draw, you can also keep Doomsayer against an aggressive deck.
This is a combo deck, so your plan is fairly simple.
1. Draw into your combo cards.
2. Stall until you have enough mana and combo cards.
3. Burn your opponent down with spell-damage boosted spells.
In a little bit more detail, play out your earlygame draw cards aiming to maximize both mana efficiency and earlygame damage to the opposing hero. This means Minion draw before Spell draw and Loot Hoarder before Novice Engineer. Coldlight Oracle is a bit unique because, while it is draw, it can also give your opponent the cards needed to stop your combo or kill you; thus, it is generally reserved for turns when you have nothing else to do and have the enemy board frozen or Ice Block up. Bloodmage Thalnos should in general be reserved as a possible damage boosting resource against control decks, however against aggro or with a bad hand, feel free to play Bloodmage Thalnos just to cycle a card.
On average, Ice Block, Frost Nova, and Blizzard will allow you to stall 4-5 turns in any game. This gives you about 6 extra draws and extra mana to play card draw. Doomsayer is a highly versatile card that can be played against turn 2 against aggro and held longer to combo with a freeze spell against a control deck's full board. Fireball is a perfectly fine stall tool as well as it costs 4 mana making it hard to use in a combo, use it to remove a threatening minion with an aura-like effect (eg: Priest of the Feast). In tough situations, using Ice Lance or Frostbolt is the correct play, but be very careful when doing this since that often decreases your chance to win the game.
As for combo, this deck is strong because there are multiple ways to win. For the most part, these amount to "fling discounted and damage boosted spells at the opposing hero". Most of the time, this is done in one turn with Fireball occasionally being used the turn prior as it is too expensive to play on the combo turn. The most reliable way to finish off your opponent is with Cult Sorcerers, Frostbolts, Ice Lances, and a single Roaring Torch for exactly 30 damage. This requires 3 mana worth of discounts otherwise known as one Sorcerer's Apprentice. However, the damage doesn't stop there since with a second Apprentice or Emperor Thaurissan along with the other Roaring Torch, any Fireballs, and the rare but possible spell damage boost from Bloomage Thalnos, you can do upwards of 50 damage in one turn. Obviously getting to this point is unlikely, but this potential alone allows for very consistent kills on or before turn 10. Always count damage when you have discounted burn in hand. There is a decent chance you can end the game.
Control/ C'thun: Your hardest matchup, keep all the burn you can using as little of it as possible for removal. Save every spell possible for use in an OTK if you can. Earlygame minions like Loot Hoarder and Novice Engineer are crucial to get in the first few points of damage.
Dragon/ Pirate/ Tempo: No deviations from the gameplan, try to protect your Ice Block if possible.
Patron: Usually you won't find out about Patron until they play the card, but if you know in advance save Frost Nova + Doomsayer or Thalnos + Blizzard for their board.
All: Stick to the gameplan and protect against damage as much as possible, even if it means being mana inefficient. Use burn fairly liberally unless you are very far behind in which case go all in for the combo.
Aggro: Same as Hunter. Be very careful of Doomhammer; this is the most likely matchup where Frostbolt/ Ice Lance is used outside of a combo.
Spell/ Fandral: Be aware of Evolve Scales, save a few more spells for the combo turn.
Beast: Stick to the gameplan, if they get too many creatures on the board for you to deal with, go all in on a combo.
Freeze: You should be favored here because of the large amount of burn. Draw and combo quickly though because you really don't want to run into the Reno variant,
Tempo: Similar to Hunter.
All: No changes from the gameplan. If your hand gets too clogged with cards, feel free to use Blizzard or Frost Nova on an empty board.
Reno: Go for the OTK.
Zoo (and Discard variants): Stick to the gameplan, go all in on combo if too far behind.
Control: Go for the OTK, they have burst healing.
Aggro: Same as Zoo.
Control: Go for the OTK.
Tempo/ Resurrect: Same as Zoo.