HTTP 418 - i.e. HTTP response code 418, "The server refuses the attempt to brew coffee with a teapot." (Which I had no idea was a thing until just now!)
My first is the first of an outrageous utterance
An 'outrageous utterance' is HYPERBOLE. Take the 'first' part of this word to get HYPER.
My second the second of appropriate circumstance
'Appropriate circumstance' is CONTEXT. Take the 'second' (i.e. end) part of this word to get TEXT.
My third's the essence of heat, why anything cools
This is TRANSFER, since things cool when heat is transferred to the environment or other objects.
My last a procedure, all about the rules
This defines PROTOCOL. These four clues together give us HYPERTEXT TRANSFER PROTOCOL, which can be abbreviated to the four letters HTTP.
Take the B's from my one
And the A's from my two
And with three's A & B's
My name I'll tell you
At this point I suspect the final number required is 418. This can be created with the help of A1Z26. Since 'my one' (first letter) is H (8), take off two "B"s (i.e. 2s) and you have 8 - 2x2 = 4. Since 'my two' (second letter) is T (20), take off two "A"s (i.e. 1s) and you have 20 - 2x1 = 18. Combine the two to get 418.
(OP points out that intended maths was simpler - taking "A's" to be the vowels AEIOU and "B's" to be the consonants (including Y), there are 4 consonants in clue 1 (HYPER), 1 vowel in clue 2 (TEXT) and 8 letters in total in clue 3 (TRANSFER), enabling us to derive '418' that way...)
The message produced in a web browser when error code HTTP 418 is generated is "The server refuses the attempt to brew coffee with a teapot."
This fits with the title, since:
Both fire and water are involved in boiling tea in a pot, and when it is ready the teapot 'roars' (or whistles). This also tallies with the dislike of coffee in hint 1...
I would love to know how exactly you can generate this specific error message! If it's just a programmer's joke then it sounds like my cup of tea...!