This is an upgraded version of my Super-Budget Mono Red Burn: http://tappedout.net/mtg-decks/super-budget-mono-red-burn/
Green is added for Destructive revelry as a way to deal with Leylines and Atarka's Command as a multi-function spell.
This deck is not budget but it isn't super expensive as it doesn't use Goblin Guides.
Sideboard:Destructive Revelry: comes in against affinity, TR0N and Leylines. It not only destroys them but it shocks it's controller also.
Dragon's Claw: These come in against other burn decks. These will keep them from burning you out first and gives a huge edge to you.
Leyline of Punishment: These are in the sideboard to bring in against things like Soul Sisters, Green lists that play Kitchen Finks and Abzan lists that run Siege Rhino.
Pyrite Spellbomb: This is a way to deal with Kor Firewalkers. It also can be used to draw a card if your hand is empty and there is no Firewalkers on the field, or can be used for the last bit of damage. If you are against another burn deck and you saw white mana from them, these should come in as many R/W burn decks have a pair of Firewalkers in the sideboard.
Searing Blood: These come in against creature heavy decks or decks that use a creature to set up a combo (with 2 toughness or less). The ability to shock a creature then bolt your opponent if that creature dies is valuable. Searing Blaze can be better in this spot, I just happened to have a pair of searing blood already and have been using them.