burn is part of the action use to get it. AKA if you have a power that says you activate as a standard and can accept burn to increase duration then it's part of that standard action. If you're doing it on something that is a free action, then it's part of that free action. I believe that increasing defense is a standard action. But yes, if you do things that cost burn you take all the burn.
There are 2 kinds of infusions, form and substance, you can apply one of each onto a blast.
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Infusions are used as part of an action. A kineticist can have one form infusion and one substance infusion active with any given blast.
Burn is taken as a cost to use wild talents or as cost to empower them. There's no action type mentioned for accepting burn though. I assume that means it's a free action. The burn cost is taken automatically when you use wild talents that cost burn.
That said, you can only take a certain amount of burn per turn, and there's a limit to how much you can take per day. The limit is 1 point of burn per turn from levels 1-5, 2 points of burn per round from levels 6-8, 3 points of burn per round from levels 9-11, and so forth.
Of course, you can reduce the burn cost of an ability by using Gather Power first. Which is a Move action.
As a GM, I personally would rule that accepting burn to improve your elemental defense is a non-provoking swift action. But there's no documentation regarding what type of action it is.