CO2 availability to phytoplankton in oceans and climate change impacts
Short Answer 
I do not believe that CO2 will become less available to phytoplankton on a global scale in the foreseeable future. Instead, I believe that rates of increasing concentrations of CO2 (and resulting chemical products) will not remain sustained at their current levels.
There are two global change phenomenon occurring that are relevant :

increasing concentrations of atmospheric CO2 and
increasing global temperature.

As CO2 concentration increases in the atmosphere, diffusion will lead to higher concentrations in ocean waters as well. The rate of diffusion is related to the relative concentrations of CO2 in both air and water.
Increasing temperatures can impact this diffusion rate as well in two ways:

Increased atmospheric temperatures result in higher surface water temperatures, which leads to less mixing of water (i.e., ocean stratification). This functionally reduces the volume into which CO2 can diffuse and therefore will slow the diffusion rate. (but not decrease the amount of dissolved CO2).

Increased temperatures often decrease solubility of gasses in water, and this decreased solubility may result in offgassing of CO2. However, decreased mixing of strata may result in less upwelling and therefore less offgassing.


As long as global ocean temperatures don't get too warm and atmospheric CO2 doesn't get too low, it seems that the above will result in a net global slowing down of CO2 absorption into oceans but not a reversal (i.e., loss of CO2). In other words, I do not believe absolute CO2 availability to phytoplankton will decrease, but that only the rate of further increased availability will decrease.

Though perhaps such decreases could be experienced as local phenomena?

 Long Answer 
It's not immediately clear to me what you're referring to. I'll try to react with some things that come to mind. My response is mostly biogeochemical in nature, so I invite others to address this physiologically if appropriate.
Are you sure you didn't misread something and it's really the declining CO32- that you read about?  (see here). Assuming that's not the case, read on...
Raising Temperatures may decrease the absorption rate of CO2
Alternately, maybe you read that as oceans warm, the water can absorb less CO2 (as in the rate of absorption will decline).
This is due to warm water not mixing as well with cooler lower water, so you end up with unmixing layers (called stratification). Eventually, the surface layer saturates with CO2 and can't absorb more from the atmosphere. It's not that this top layer of water has less CO2, but rather that it gets so saturated that it can't absorb any more (causing the rate of absorption to decline).

Under less warm conditions, the &quot;layers&quot; of water aren't so different in temperature, and so more mixing of layers can happen. As a result, all that CO2 that is being absorbed at the surface level can be exchanged to lower layers of water where it will be stored longer. In other words, with cooler surface temperatures (and therefore greater mixing of layers), the volume of water that can hold CO2 is no longer just the surface, but a much &quot;thicker&quot; layer of water. Result: more CO2 absorbed.

So... raising temps can cause less mixing of water due to more stratification (layering), which results in less water in the ocean available to absorb and hold the CO2. This means that as atmospheric CO2 continues to increase, the non-mixing surface layer of ocean water (which will become saturated with CO2 at some point) won't be able to keep up with more and more and more CO2 in the air. As a result, the ocean will decline (and eventually potentially fail) in its ability  to &quot;buffer&quot; the ever increasing CO2 in the air. This would mean that the rate of CO2 in the air will start increasing more rapidly (since less and less of it is being absorbed by the oceans).
As for the phytoplankton (which are in this top layer of water), this stratification will not directly result in less CO2 availability to them. As atmospheric CO2 increases, so will the amount in this top layer of water. The rate of increase in CO2 concentration will just slow until a saturation point is reached, but absolute levels will not decline.

Though, note, however, that phytoplankton tend to thrive in areas of high nutrients (i.e., upwelling zones). Less mixing of waters will decrease the upwelling of nutrient-rich, cooler subsurface water. So less mixing (i.e., more stratification) likely would lead to declines in phytoplankton abundance due to decreased nutrients availability. (See here). Perhaps this decreased nutrient availability (which would include loss of carbon sources) is related to what you're referring to?

Read here for some more thoughts: https://earthobservatory.nasa.gov/features/OceanCarbon
Raising Temperatures may decrease CO2 solubility
However, given all this, the solubility of CO2 in water does decline with increasing temperature (see here for raw data). This suggests that some rise in global temps may impact CO2 concentration in ocean waters.

The cause for this decreased solubility with increased temperature is due to the imbalance of an equilibrium state of free energy required for dissolving/off-gassing gasses in/out of water. Since the dissolution of CO2 into water is an exothermic reaction (i.e., one that releases heat), any addition of heat will lead to a greater favoring of the opposite endothermic reaction (release of dissolved gasses in this case). In more chemical terms, added heat to the solution provides energy to overcome attractive forces between the CO2 and the solvent (i.e., water) molecules. The result is decreased solubility. See here and here for further explanation.
(The partial pressure of gasses are also impacted with increased temperatures, so investigating Henry's Law might also be pertinent to your research. (This link I found quickly explains a bit of the phys/chem).)

However, to further complicate all of this, not all of the CO2 absorbed by ocean water remains as carbon dioxide. Much of it undergoes reactions to become carbonic acid, carbonate, etc. (see here). This means that the equilibrium state is more complex to model or understand vs simply examining these effects on CO2 in water in a controlled lab setting.



This solubility fact, in combination with the mixing of layers above, are why polar waters tend to absorb more CO2 and equatorial waters tend to give off more CO2
From http://www.carboeurope.org/education/CS_Materials/CO2solubility.pdf:

In the world oceans, the northern Atlantic and the southern Oceans act as major sinks of CO2 because they are colder. Adding to this, cold water is denser than warm water, causing it to sink. The carbon dioxide taken up in the surface can be effectively transported to the deeper waters by convection hence the CO2 is stored in the bottom layers of the ocean.


On the other hand, warm equatorial waters tend to release CO2 into the atmosphere. In these regions upwelling of CO2-rich deep waters occurs. When the water reaches the surface, it is warmed decreasing gas solubility leading to the degassing of CO2 .


However, note that if there is greater stratification of water (i.e., less mixing) as described above, such upwelling could actually decline. This would result in trapping more CO2 in cooler waters instead of warming those wters as they rise to the surface and offgassing that CO2 back into the atmosphere. (We won't even begin to discuss how ocean currents in general will be impacted by changes in temperature...).

To add even more complication, as waters warm, ice will melt leading to (at least temporary) increased mixing of ocean waters!! (see here). If temperature decline from ice melt is great enough to combat the increased surface temperature, then perhaps for some period greater mixing and offgassing could occur.


End results
What does this all mean for climate change?
Well, as atmospheric CO2 increases, it's going to make the oceans more acidic (due to carbonic acid). In other words, if we kept temperature constant, more atmospheric CO2 means more ocean CO2. (You can see that from my previous two links -- again, here (showing ocean CO2 concentration on Y axis) and here).
However, as the earth warms up, water temps will warm up. This means less mixing and therefore less net CO2 absorption. It also means that the warmest surface waters will not hold as much CO2 due to its reduced solubility, which will result in greater &quot;off gassing&quot; of some CO2. The combination of these two things means essentially that the rate of CO2 increase in the atmosphere will likely accelerate.
But what is the net impact on oceans? (and therefore on plankton)
Well, in regards to the oceans themselves, there is probably some calculable &quot;tipping point&quot;  in which the higher concentration of CO2 in the atmosphere (and therefore greater diffusion pressure) doesn't outweigh the reduced solubility of CO2 (due to higher temps and less mixing). At whatever temperature that tipping point occurs, then I assume that the CO2 in the oceans can/will decline instead of increase. In reality, though, gasses are constantly moving back and forth between atmosphere and ocean, so it's unlikely the ocean water will get so warm that there is a net movement out of the water, but probably just a decrease in the rate of further absorption.
On a more local level, if the phytoplankton are living in an especially warm region of ocean, it is possible that off-gassing of CO2 increases in those regions due to the decreased solubility as described before. However, again, if the concentration of CO2 in the atmosphere and the temperature of the atmosphere both keep increasing, then these effects would (at least partially) negate the increased offgassing from reduced CO2 solubility. Again, modelling when this would occur is much too complicated for me to figure out for this post, but I'm sure we have experts working on calculating this &quot;tipping point&quot; as we speak.

Again, the bigger issue for the phytoplankton is probably the reduced upwelling of nutrient-rich water due to greater climate-mediated stratification of the ocean column.

Final Thoughts
Given all of this explanation, ice-core research by Pedro, Rasmussen and van Ommen (2012)  (summarized here) suggests that after only a few hundred years of warming, CO2 concentrations seem to decline in ice cores. This may suggest that perhaps a net release of CO2 is possible within a hundred years or so. Again, this &quot;tipping point&quot; is likely calculable, but it's too complicated for me to figure out for a stackexchange post ;p.
Further readings:
Basu, S. and Mackey, K.R., 2018. Phytoplankton as key mediators of the biological carbon pump: Their responses to a changing climate. Sustainability, 10(3), p.869.
Hülse et al. (2017) Understanding the causes and consequences of past marine carbon cycling variability through models. Earth-science reviews, 171, pp.349-382.