According to the BBC, some researchers have analysed the size of the plates in depictions of the Last Supper between 1000AD and either the 1700s or 1900AD (the article gives both as dates for the most recent painting analysed).
This research is not, as might be expected, found in an interesting journal of art history, but in the International Journal of Obesity. I don't have access to the full text of the research, so I can't tell how good it is. However, the news article is not hopeful.
The Guardian article gives slightly more details, including the source they used to find the paintings (which are mostly Western European and Russian in origin, judging by the Amazon's index).
So, flaws in at least the reporting, and possibly the research too:
Firstly, the differences in style between eras of European painting are ignored. Pre-renaissance painting was largely symbolic, for instance, and the pursuit of a "realistic" style has varied considerably over the centuries. The Last Supper, as one of the major events of Christianity, has a lot of associated symbolism. That starts to make comparisons based on a simple head:plate ratio less valid (though not as invalid as they would be from a larger selection).
Neither of the researchers is an art historian by speciality (Reuters describes what they do specialise in), but without the full text I can't tell if they asked someone who was or not.
Secondly, the use of plates may have changed over that time. Was every plate depicted a plate of comparable function? Without the source material and a better knowledge of the technology I can't tell. There is a risk, though, of comparing side plates to main plates. Similarly, changes in numbers of courses may have had an effect - if the number of courses being eaten increases, the average plate size for a course will probably decrease, and vice versa.
Thirdly, the reporting seems to be linking this to "portion sizes are getting bigger so there's an obesity crisis!!!". Since the phrase "super-sized" appears in all three news articles, despite them otherwise being very different in wording, I'm guessing this was introduced either by the press release or by Brian Wansink.
The major problem with this is that the range 1000AD to 1900AD was not characterised by any sort of obesity crisis whatsoever. It was characterised by major famines, general malnutrition, and other problems related to lack of food (especially for people outside the nobility, and especially towards the start of the period).
As farming and storage technology improved, and as European nations invaded large parts of the rest of the world to bring increased prosperity at their expense, it probably is true that over those 900 years the amount of food available to Europeans generally increased. This may have had a general effect on the size of plates and the amount of food on them in artwork - that's entirely plausible.
The conclusion, however, that some people seem to be drawing, is entirely unfounded. From the BBC article:
Charlene Shoneye, an obesity dietician for the charity Weight Concern, said: "I'm really not surprised by these findings because the size of our plates and food portions has increased.
"Twenty years ago, for example, most crisps used to come in packs that were 20g. Now they are 30g, 50g or even 60g, and we are still eating the whole pack.
"This super-sizing has changed our perception of normal."
But she said it was not too late to reverse the trend and that individuals, society and the food industry should look to smaller portions.
So, this person is not surprised at a general increasing trend between 1000AD and 1900AD because of a separate trend between 1990AD and 2010AD1? This is fairly typical of some of the "science" behind the "obesity epidemic", but not being able to tell the difference between a trend based on the slow eradication of malnutrition and a trend based on marketing decisions is not a good sign.
Widespread malnutrition probably would stop most people from being above the government-mandated weights, but it's not a great policy idea. The whole "obesity epidemic" panic, though, seems to be predicated on a "wasn't everyone healthier when they were borderline-malnourished" idea - with "so let's put them on diets to induce this state" the obvious conclusion.
It's a conclusion that can only be drawn when you have the privilege not to be able to personally or culturally2 remember widespread and unavoidable malnutrition.
1 Further on the crisps point - yes, the packs do vary in size (you can even get 100g and 150g packs of some brands), but the cost varies with the size. A 50g bag of crisps is more expensive than a 30g bag of crisps. The 30g bag of crisps is probably more expensive (after correcting for general crisp-related inflation) than the 20g bags were.
2 There are people, and groups of people, of course, even within the Western European and Northern American countries currently worrying about the "obesity epidemic", who don't have enough food. Society and privilege make them sufficiently invisible to many "obesity researchers", though.