Nocturne, Horrifying

No — Burn only puts cards into your trash; it never looks at or reveals anything, and Nocturne's ability only cares about those two specific actions. Minefield's printed ability works the same way, and this won't change even if its wording is later updated to explicitly read Burn 2.

Nocturne's ability reads "As you look at or reveal me from the top of your deck, you may banish me. If you do, you may play me for ." That phrasing ties the permission to the specific action of looking at or revealing Nocturne — the same construction the rules use elsewhere to key an effect to one exact moment rather than a general state, such as choices made "As I am played."^[355.1]

Reveal is a defined action, and a player may only reveal cards when a game effect specifically instructs them to.^{[424.1][424.2][424.2.a]} "Look" doesn't have its own numbered definition yet, but it's treated the same way — the main deck is secret information specifically because it "cannot be looked through or revealed unless instructed by a card effect."^{[103.2.e][103.2.e.1]}

Burn does neither of these — it puts cards from the top of your main deck directly into your trash, with no instruction to look at or reveal them.^[431.1.b] Minefield's printed ability, "When you conquer here, put the top 2 cards of your Main Deck into your trash," does exactly the same thing — it's already functionally identical to Burn 2, so an errata formalizing it as such wouldn't add a look or reveal instruction that isn't already there. Either way, those cards only become visible because they land in the trash, a public zone by default,^[108.2.d] not because anyone looked at or revealed them first.

Since neither instructs anyone to look at or reveal Nocturne, its ability is never triggered by Burn or by Minefield.