Shepherd tells Brianna that the volcano they're on, Trollsvatn, is where Jules Verne's fictional explorers went underground in "Journey to the Center of the Earth". It's not. The passage Verne used was in a volcano under the glacier Snaefellsjokull.
A voiceover from a news broadcast tells of the city of Rome being threatened by eruptions of Mount Vesuvius and Mount Etna. Vesuvius is in the vicinity of Naples, over 100 miles from Rome; Etna is on the island of Sicily, over 200 miles from Rome and separated from the Italian mainland by the Straits of Messina. Neither mountain is in a position to threaten Rome, even in the case of a severe eruption. The only possible reason for mentioning them is 1: they're volcanoes and 2: they're in Italy.
At a university, a bell rings to end the class. This is only typical of classes in K-12 schools in the U.S. College courses vary in lengths of time usually from 30 minutes to 2 hours, & thus, not all classes begin or end at arbitrary times.
In the opening scene, Jeff's first line is, "Tell me again why we're sitting on top of this rock called Trollsvatn?" The Icelandic word "vatn" means "lake". Since the volcano they are surveying is a mountain, it would be more likely to be named "Trollsfell" ("fell" means "hill" or "mountain", as in the volcano Eldfell). However, most Icelandic volcanoes have individual names, often taken from the nearest landform, be it glacier, headland or other formation. There is one volcanic area in Iceland with the word "vatn" in its name, but it refers to a volcano with its magma chamber and vent under a series of frozen-over lakes at the edge of a glacier.
Peter Shepherd tells Brianna Chapman she has to buy her own plane ticket if she wants to travel to Iceland with his team. The team is flying on a chartered jet which does not require an individual to purchase a ticket. This would only be needed if they were traveling on a commercial flight.
The submarines are launching nuclear armed torpedoes which have been part of various nations' arsenals since the 1950s. However, the detonations often appear to be conventional. The submarines are far too close to the explosions to survive an underwater nuclear detonation of even few kilotons.
It makes no sense that magma would drip from above into the lava tube. Elsewhere on the same level, they find a magma chamber that is rising. The magma would spread into the tunnels, including the lava tube, before it could get into an area above the lava tube. The lava tube would have been filled with magma before any other magma could break through the ceiling into it.
On board the British submarine, a flag is painted on the periscope housing. This flag, red background with crossed swords a lion and a crown is that used to represent the British Army - who do not operate submarines.