[Quoth]

If you are the author, you can decide if Emphasised or Italic is correct, otherwise it's very hard to decide.

so
“<em>No! No! No!</em>”
or
“<i>No! No! No!</i>”

But really neither is needed with an exclamation mark. Some might use caps
“NO! NO! NO!” he shouted.

I've seen people use bold (or strong), but really I think that's poor style, because Bold is for headings. It looks out of place in dialogue.

Obviously text books can use bold or italics to differentiate a word, and might justifiably use <em> or <strong>. However doing any styling, sematic or raw, in dialogue seems clumsy. Common enough in Georgian and Regency novels and gradually died out in the Victorian era.

Italic style is more often not used for emphasis, with telepathy or mindspeech etc you might use:
<i>No</i>! <i>No</i>! <i>No</i>!
and always omit ther dialogue quotes because the italic alone indicates it's dialogue without speech.
Italics used to be used for internal dialogue (the thoughts of a character) with quotes, but most style guides now prefer you indicate with an I/he/she thought pseudo tag. No quotes or italics.

<em> should ONLY be used for emphasis. I've seen entire books by the House of Random Penguins where the author is dead and they simply use <em> and <strong> where the paper print had italics and bold, and that's simply delusional. Maybe sometimes you can tell the semantic intention from reading the paper text of a dead author, but mostly you can't.

Use the semantic flavoured tags if you want and think it's justified. Mostly if it's spoken dialogue the enclosing quotes should be normal body style, i.e. outside the <em> or <i> or <strong> or <b>.

Really authors should decide on semantics, not purely typesetters/formatters etc, because how can they know the author's intention?