As the number of ocean dead zones increases, is the world's primary source of atmospheric oxygen (phytoplankton) at risk?
Yes, it is, though probably not immediately. Ocean and atmosphere deoxygenation is a thing, and many publications are out.
In a larger context, deoxygenation is indeed inevitable even if we ignore anthropogenic warming. Increasing solar flux alone (the sun get's hotter over its lifespan) could reduce the atmospheric oxygen to 1% of present day level within around a billion years, says this publication.
This publication estimates that almost 80% of ocean deoxygenation under current day CO2 levels have not been realized yet, but they say that oceanic surface deoxygenation will stop once CO2 emissions stop, while the deep ocean oxygen levels will continue to decrease.
A global warming by 6°C (which is at the upper end of current projections until the end of this century) could tip the oxygen production by phytoplankton, cancelling the ocean's role as a main oxygen producer.
Citing from the abstract:

[...] If, in the course of time, the oxygen production rate becomes too low or too high, the system’s dynamics changes abruptly, resulting in the oxygen depletion and plankton extinction. Our results indicate that the depletion of atmospheric oxygen on global scale (which, if happens, obviously can kill most of life on Earth) is another possible catastrophic consequence of the global warming [...]