The word "fine" has many meanings, some of which work well with "very" and some which don't. The uses of "fine" as a noun or verb definitely don't take very, but I don't think that was in doubt. I'll focus on the adjectives, since those could describe a person (thus might be valid in the phrase "I'm fine").
Going with the definitions from Merriam-Webster, even just as an adjective there is a wide range of meanings.
Definitions 2a and 3 roughly cluster the meanings of fine which are "small", and "very" does work with these: a "fine wire" is a thin wire, and a "very fine wire" would be even thinner.
Definitions 1, 2b, 4, 5, and 7 are all based on fine meaning "good", and "very" works with these as well: a "fine day" is a good day, a "very fine day" is even better.
However, in my experience the phrase "I'm fine" always uses fine as an adjective with definition 6a:
well or healthy : not sick or injured
In particular, I would describe it as an ambivalent term: not significantly "good", but also not significantly "bad". Such a term doesn't really work well with the amplification that "very" brings, especially since it's usually thrown about in a non-committal way. For this definition, I would not use "very".
To put things a different way, the phrase "I'm fine" is intentionally vague, while the word "very" adds specificity. If you want to tell someone about yourself in more detail then you wouldn't be using "fine" in the first place and should just use a less-vague word.
As a side note: the typical meaning when describing another person as "very fine" is "that person is extremely attractive". If you intended to describe yourself as very attractive then "I'm very fine" could be valid, though I would recommend the phrase "I'm so fine" instead.
(Lastly, a disclaimer: I only know American English, and could not even say whether a difference exists for this subject between American and British English, let alone what that difference would be.)