What is the non-local impact of a large-scale destruction of coral reefs?
Disclaimer: the question is too broad and speculative to some extent, and thus less likely to get complete answer (at least in a single post).

As is already known, thousands (or millions) of species are dependent on coral reefs for their survival and are likely to get extinct (unless they adapt) because of destruction of coral reefs. A short list includes fishes, turtles, sharks, eels, crabs, shrimps, urchins, sponges, algae, and so on1. Now, telling which of them will be immediately affected will be largely speculation, yet the first 5 species to lose the race, as per a speculation, are2:


Butterfly Fish: Species of fish that are obligate corallivores, such as many butterfly fishes (Chaetodontidae), feed exclusively on coral polyps. Obviously, survival would not look good for these fish if their food source disappeared.  Like parrotfish, butterflyfish assist in keeping algae from smothering corals.


Spiny Lobsters: Spiny lobsters rely on coral reefs for protection, especially during their vulnerable molting episodes. These lobsters have a significant role in maintaining a balanced ecosystem. Like sea otters, spiny lobsters are predators of sea urchins, which feed on kelp forests and can destroy them if populations are not predator-controlled. Kelp is important for many reasons, including those listed on OGP’s kelp recipes.


Whales (and dolphins): Many scientific studies have substantiated that dolphins and whales have extreme emotional intelligence. Sadly, their intellect won’t be enough to help these animals if coral reefs are destroyed. Dolphins and most whales (cetaceans) are predatory animals and they are all carnivorous. It’s a fish eat fish world and many fish survive by consuming fish that rely on coral reef habitats. Every species lost will affect another…and another…and many others. So, these come under the indirect adverse effects of coral depletion, since they themselves don't rely on corals.


Whale Sharks: Whale sharks are an important indicator of marine productivity and have a role in the conservation of other marine organisms. Although there has not been adequate research to understand why whale sharks are dependent on coral reefs, it has been shown that decreases in whale shark populations in the late 1980’s and early 1990’s may have been linked to increased rates of coral destruction.


Hawksbill Sea Turtles: Although all sea turtles would be affected by the loss of corals, Hawksbill sea turtles have become critically endangered species and are highly dependent on coral reefs for their food sources, mainly consisting of sponges. Sea turtles play significant roles in their ecosystems in that they help with nutrient cycling from ocean to land, maintaining healthy sea grass beds, and balancing food webs.




Coral reef fishes are in general rather sedentary and territoriality and home range behavior patterns are highly developed. However, many species migrate, often relatively long distances compared to body size and often with spectacular precision. Migrations in reef fishes may be associated with3: 


Life history: movements of 
planktonic larval stages to reefs or movements of juveniles from nursery areas to reefs
Seasons: precisely timed spawning aggregations drawing fishes to particular locations from wide areas of the reef
Diel patterns: movements to and from feeding or resting areas associated with dawn and dusk. 


Another species, which is to be affected, though not immediately, by destruction of corals, is humans(!). As per experts4, coral depletion will bring threat to not only income and economic stability, but also to survival by causing depletion of food and increased risk of coastal disasters, which are likely to kill millions.

Coral reefs also serve many purposes, a few of which are5:


protect coastlines from the damaging effects of wave action and tropical storms
provide habitats and shelter for many marine organisms
are the source of nitrogen and other essential nutrients for marine food chains
assist in carbon and nitrogen fixing
help with nutrient recycling.


Finally, I suppose most of the species will be affected because of depletion of source of food rather than more-than-sufficient amount of something. Also, I was unable to find any simulations of the global consequences of destruction of coral reefs. I hope this much would help!

EDIT: I did some online research, and found lack of research about this topic. But I found a paper regarding relationship between seagrass, mangroves and coral reefs6 regarding life cycle of migratory fishes. From the same:


  Coral reefs showed the highest mean fish species richness and were dominated by adult fish, while juvenile fish characterized seagrass beds and mangrove sites...Our results suggest that connectivity of seagrass, mangrove, and coral reef sites at a species and site levels, should be taken into consideration when implementing policy and conservation practices.


It becomes evident from this research that depletion of coral reefs would certainly affect life cycles of these fishes, which include Scarus iseri (Striped Parrotfish) and Lutjanus apodus (Schoolmaster Snapper), which will, in turn, affect biodiversity in seagrass beds and mangroves, leading to further break in food chain in more diverse areas.

Some researches have also suggested that seabirds might also be on a high rank in this list. Seabirds survive by eating fishes, and studies have shown that increasing water temperatures halt their growth while higher sea levels cause them problems in nesting. Now, it has also been suggested that due to coral bleaching, many fish species which live on corals might either die or move to other places, because of which seabirds could also face survival crisis7.

Another point regarding your demand I'm interested in effects which are not obvious and/or have not been broadly publicized, specifically not been broadly publicized, is that it has now become difficult to find such papers. This topic (climate change) has become so much publicized now that all related papers get attention immediately. So, it is quite difficult to find environmental research papers about climate change which hasn't gained much attention. Yet, I'll keeping on adding more research papers as I find some more.