Why "Having chlorophyll without photosynthesis is actually very dangerous" and "like living with a bomb"?
Chlorophyll absorbs photons (light). The energy in the photon extracts an electron from a molecule of water. Electron transfer creates intermediate superoxide and hydroxyl radicals from the oxygen and hydrogen from the donor water molecule. 

In normal photosynthesis, these radicals are quickly used to power the reduction of NADP to NADPH and the synthesis of ATP from ADP. NADPH and ATP in turn power the synthesis of sugars from carbon dioxide and water, via the Calvin cycle.

These radicals are highly reactive. If they hang around and don't get used properly in redox reactions, they will just as happily attack DNA, proteins, and structural lipids within the cell, and are therefore dangerous. In normal plant cells that get too much sun, for instance, free radicals can build up and cause cell damage. 

My guess is that the author's "bomb" is made up of higher concentrations of these radicals within a cell with no apparent machinery to perform the downstream (photosynthetic) chemical reactions needed to consume them safely.

Edit

I skimmed the (sadly, paywalled) paper, and it sounds like my guess was right, that it is indeed these radicals that are the danger from having chlorophyll, with no light-independent (Calvin cycle) mediated reactions to safely consume the energy in them:


  Chlorophyll itself has no natural biological function outside of
  photosynthesis, so if photosystems are indeed absent, corallicolids
  must have evolved a novel use for either chlorophyll or its closely
  related precursors or derivatives. However, these molecules generally
  function in light harvesting, which would be destructive to cellular
  integrity without the coupling of the resulting high-energy compounds
  to photosynthesis. Other possibilities are functions in light sensing,
  photo-quenching or the regulation of haem synthesis, but these too
  leave open the question of what the cell would do with the highenergy end products. 


What's not clear to me is that the genes that help generate chlorophyll are expressed, but the cells are unpigmented. I don't see any explanation where the chlorophyll and associated proteins are localized in the cell — seems like a missing part of the paper, or I missed that part when skimming. Or perhaps the organism has evolved interesting and novel ways to manage the damage caused by these oxygen radicals, or has other mechanisms for consuming them, yet to be identified. 

Should motivate further research, especially if these organisms share ancestry with malaria and toxoplasmosis — there might be something interesting to learn that would help with eliminating these diseases. I imagine a biochemical "radical bomb" could be very handy for destroying parasites, if there was a way to expose them to light or some other source of photons; perhaps some drug therapies could target the relevant genes and induce the parasite to destroy itself. Interesting paper.