I feel like it should be alright, but thought I'd get your feelings on whether it's appropriate to employ the term "got burned" if no fire was involved, specifically as it pertains to injury to a living thing, or if I am precluded from having been burned by hot water, and have to employ scalded or cooked or some other non flame related term?
This definition here seems a good example of why your usage is okay:
to injure or damage by or as if by exposure to fire, heat, or radiation
burn v "burned my skin with the acid." (TFD)
3. To damage or injure by fire, heat, radiation, electricity, or a caustic agent: burned the toast; burned my skin with the acid.
So, yes, I guess you could say that.
Burns are injuries to tissues caused by heat, friction, electricity, radiation, or chemicals.
The source of this heat may be the sun (causing a sunburn), hot liquids, steam, fire, electricity, friction (causing rug burns and rope burns), and chemicals (causing a caustic burn upon contact).
(src: TFD Med.)