Arc Fatigue: The first several chapters are just Axel and the Professor's journey from Germany to Iceland. They can be skipped without missing anything. These chapters are boring enough in the original, but in the most well-known English translation, they become downright unbearable.
From the 2012 Mysterious Island movie, the whole "Pop the Pecs" scene would certainly qualify as one of these.
One of the English translations of the book added a dream sequence where Axel (or "Harry", as he is renamed) witnesses a battle between a giant ape and crocodile and is then attacked by the ape before waking up. It's a completely pointless scene that only succeeds in making the reader further lament that the story doesn't feature more prehistoric creatures.
Cargo Ship: Hans appears to have an odd relationship with his duck.
Critical Research Failure: The last name of the character Hannah Ásgeirsson in the 2008 film should by all rights be "Ásgeirsdóttir" (meaning "daughter of Ásger", as opposed to "son of Ásgeir"). Icelanders don't pass their last names onto their children, so Icelandic women with last names ending on "son" are pretty much unheard of.
It gets worse: Hannah's father's name was Sigurbjörn Ásgeirsson, so she should have the last name Sigurbjörndöttir.
Fridge Logic: The 2012 film is jam packed with the stuff. Between the fact that there's apparently a category 5 hurricane just hanging around somewhere off the coast of Palau (and nobody on the island seems to care too much about that,) and the fact that there's all sorts of satellites overhead that might be interested in taking a look at what's going on in there, it doesn't seem like the Mysterious Island would stay mysterious for all that long, would it? Oh, and you'd also think one or two of those satellites would be equipped with spectrometers that might take notice of a giant gold-spewing volcano...
Conventional ham radio does not use satellites; signals may however be skipped off the ionosphere.
How does one steer a giant untrained bee?
Maps are optional features in most books.
Verne's Nautilus did not have torpedoes (Nemo sinks his shipping by ramming), and Professor Arronaux was light on the details of the submarine's controls. And batteries do not work like that.
Sean is in high school, and clams to have straight A's. Why doesn't he know about latitude and longitude cordinates, then?
How are there living land-animals on an island that spends long stretches of time underwater?
Narm: Pat Boone's Alec shouting out "Hi-ai!" to alert his fellow spelunkers, just for how awkward the antiquated expression sounds to a modern ear.
Non-Singing Voice: Averted in the 2012 movie; that really is Dwayne Johnson singing "What A Wonderful World" with new lyrics (and again over the end credits, with the proper ones).
The Other Darrin: Sean's mother was played by Jane Wheeler in the 2008 film. In 2012, it's Kristen Davis.
Special Effects Failure: Not bad for the time (1959), except for the scene where they're carried up the lava tube. The beloved slurpasaur makes an appearance, looking much more realistic in its movements than any rubber monster.
The 2008 film has special effects that are very often hit and miss.
Tear Jerker: In the 2008 movie, the scene where the characters bury and say goodbye to Max. The entire thing, but especially the journal entry read by Trevor, that Max wrote on Sean's birthday.
From the earlier movie, the scene where Hans wakes up to find out that the Token Evil Teammate has butchered and eaten his beloved duck, arguably what pushes him over the Moral Event Horizon, though he gets his comeuppance almost immediately afterward.