I understand why they would turn yellow, but how can they turn back to green when keeping them in the dark? They absorb carbon dioxide passively somehow?
Potassium trioxalato ferrate (III) is photosensitive, i.e. it decomposes when exposed to light, reforming iron (II) oxalate.
https://www.dimanregional.org/cms/lib/MA01929605/Centricity/...
I believe that the reason it turns green again in the dark is because the oxygen of the air oxidizes the iron back to the iron (III) form. This proceeds easily if there is excess oxalic acid. In the absence of excess oxalic acid, this could still happen via a disproportionation sort of reaction that turns some of the iron (II) oxalate into insoluble iron crud (mixed oxides/hydroxides) and frees its complexed oxalate to reform the iron (III) complex.
You could test to see if it's oxygen from the air by exposing some of the solution to sunlight until it turns yellow, then storing half in a capped container with no head space for air, and the other half in an open topped container. Does the solution in the open topped container turn green faster? Does the solution in the closed container turn green again at all?
If there is excess oxalic acid present, then the extra oxalate ions can form ferrioxalate ions in the absence of light, causing the solution to revert to green.
UV induced fluorescence would have been cooler tho