These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Accidental Innuendo: It was noted even at the time it was published that some of the symbolism and word choices were a bit suggestive, which Melville apparently didn't notice until later pointed out to him.
Crazy Awesome: Ahab. It is because of his insane obsession with Moby Dick that he created charts accurately mapping the annual movements of the whales, allowing them to follow the herd, kill as they go and fill their hold in record time. His insanity also wins over his crew and makes them all (save for Starbuck) extremely loyal and invested in the hunt for Moby Dick.
To modern readers, the three harpooneers can come across as caricatures of Africans (Daggoo), Native Americans (Tashtego) and Polynesians (Queequeg), respectively. However, Melville makes them all sympathetic characters and Ishmael frequently talks about how they're Not So Different from white men.
"Better to sleep next to a sober cannibal than a drunken Christian."
Through most of Moby-Dick, the sperm whale was a monster, the legendary leviathan to be hunted down and killed for its oil and spermaceti. Melville admonshed people not to burn sperm-candles or lamp oil recklessly, not because he wished to spare the whales' lives, but because so many human sailors died every year on whaling expeditions. In the modern world, just about every species of whale is endangered, and whaling was one of the main reasons for their dangerously low numbers in the wild; "save the whales" is a rallying cry more people support than oppose and thus people are more likely now to cheer on Moby defending himself. Yet at the time the story was written, whale populations were much larger, and nothing was known of whalesong or other such indicators of cetacean intelligence.
Science Marches On: While the author was very knowledgeable about cetology, some "facts" he used have since been proven to be inaccurate.
"Be it known that, waiving all argument, I take the good old fashioned ground that the whale is a fish."
More of a case of definitions march on. "Fish" originally just meant "animal that lives exclusively in water". Melville recognises that whales are warm-blooded, breathe air, and bear live young, but just doesn't think that a sufficient reason to redefine what "fish" means.
He also mentions phrenology and physiognomy, both now considered pseudosciences.
Chapter 105 poo-poohs the notion that whaling might endanger the whale population.
Whether or not you approve of whaling, it is more controversial today than it was when Moby Dick was published, what with environmentalism and concern for endangered species.
The story does explicitly discuss whether Man could hunt whales to extinction. Already their numbers are diminishing, but the author's opinion is that they'll avoid Man by swimming North to the icy oceans, and so will always be able to evade extermination. It's an opinion based on flawed biology; given better science it could have gone the other way. Even so, Melville gives consideration to it.
This work in particular took several decades to attain the critical status it enjoys today. Melville was previously a successful author of travel books that are forgotten today; after the failure of Moby Dick, his career declined. As a literary critic noted, he's probably the only writer in history to be ruined by his one masterpiece.
In the Reader's Digest: World's Best Reading edition, Thomas Fleming states in the Afterword that critics scoffed at the idea of someone going as far as Ahab did, and everyone around simply obeying...until they lived through World War I.