you take burn whenever it says to. So if you have an ability like celerity that says you can accept 1 burn to make it last longer if you accept that burn you take the burn. Or if you want to use an infusion like the extended range form infusion, that costs one burn to use it with your blast. But since it's a blast you can reduce the cost by gathering energy as a move action to make it free again. Also your defense will power up if you spend burn on it.
A lot of things cost burn. Some things reduce burn costs. Infusion specialization can reduce infusion burn costs per appropriate level. Gather power can reduce the burn cost of infusions, metakinesis, and/or composite blasts. Only the buffer can reduce the burn cost of a utility wild talent.
So basics of burn:
Burn is a pool of something that mostly hurts you. For every point of burn you have, you take 1 point of nonlethal damage/level. Basically, in practice, just pretend that, for the sole purposes of HP, each point of burn reduces your CON but 2 points. And this stays all day until you get a good night's sleep (it expressly forbids methods of removing the nonlethal damage)
This isn't necessarily a problem with a small bit of burn, since you are relatively unique in the game- you are basically a CON based spell caster. So you have more than enough room to soak some of this up. If you have 18 CON, and you have 2 burn, then in practice it is like you are running around with a more average 14 CON.
The class revolves heavily around you using this burn. First, it powers your big shiny powers, and secondly, it powers your stat and attack/damage booster. With 'elemental overflow', the sting of burn is reduced a bit
For example, going from 2 burn to 3 is basically only POSITIVE, since you get a +2 to two of your physical stats; since one of those goes into CON, obviously, you gain enough that going from 2 to 3 do that you suffer no loss of effective health. Instead, you just get a free +2 to a stat, as well as the nonhealth benefits of more CON for this class (+1 to DCs, +1 damage on blasts, etc). Elemental overflow also gives you bonuses to attack and damage (to a limit, as indicated by the table). It adds burn directly to your attack roll, and adds burn twice to most damage rolls.
Since you don't get enhancement bonuses (no actual weapon), elemental overflow is balanced to put you roughly on par with other 3/4 BAB classes with enhanced weapons. So it is hard to avoid burn entirely, since it helps alot.
But generally, msot of the class's mechanics are built around reducing the cost of burn on your blasts, so that eventually you can just spam weird stuff. Your utility and defense talents don't get reductions though (so you either suck up the burn, or just go without using it). About half of the defense talents are often good enough that it is worth spending burn, so that you can get you attacks started off and have a bit of defense to counteract the burn (I love earth users, since at level 6, I can grab 3 burn and thus get DR/6, +3 attack, and +5 damage- enough that the barbarian is a bit jealous. More than worth the lost HP).