I'm going to guess that the stories of toxicity of dragon meat is largely modern, as I can't think of stories from the traditional canons where this is a story element.
It may derive from the story of Nessus the Centaur, Hercules and his wife Deianeira. The story can be found in Apollodorus, (Book 2, Chapter 7, Sections 6 & 7), although the Hyginus is more specific:
Nessus, son of Ixion and Nubes, a centaur, was asked by Dejanira to carry her across the river Evenus, but as he was carrying her, in the very river he tried to ravish her. When Hercules came there, and Dejanira implored his aid, he pierced Nessus with his arrows. As he died, Nessus, knowing how poisonous the arrows were, since they had been dipped in the gall of the Lernaean Hydra, drew out some of his blood and gave it to Dejanira, telling her it was a love-charm. If she wanted her husband not to desert her, she should have his garments smeared with this blood. Dejanira, believing him, kept it carefully preserved.
SOURCE: Pseudo-Hyginus, Fabulae, 34 (trans. Grant)
and the Ovid even moreso:
He shot an arrow through the centaur's back,
so that the keen barb was exposed beyond
his bleeding breast. He tore it from both wounds,
and life-blood spurted instantly, mixed with
the deadly poison of Lernaean hydra.
This Nessus caught, and muttering, “I shall not
die unavenged”, he gave his tunic, soaked
with blood to Deianira as a gift;
and said, “Keep this to strengthen waning love.”
SOURCE: Ovid, Metamorphosis, 9.98
WESTERN vs. EASTERN DRAGONS
In the west, dragons are associated with evil and fire, deriving from Greek myth and sagas such as the Volsungs and Beowulf, and extended in Tolkien.
In the East, China in particular, dragons are symbols of good luck and associated with water. (Often their role in the universal structure is to bring rain, the greatest boon to an agricultural society. i.e. rain = good fortune).
This is the idea the object are imbued with properties, and "like begets like".
If we analyse the principles of thought on which magic is based, they will probably be found to resolve themselves into two: first, that like produces like, or that an effect resembles its cause; and, second, that things which have once been in contact with each other continue to act on each other at a distance after the physical contact has been severed.
SOURCE: Frazer, The Golden Bough III.1
In the west, eating dragon flesh would produce negative effects, mirroring the poisonous and destructive nature of dragons.
In the east, consuming dragon bones would produce positive effects, mirroring the dragons' association with good fortune.
It's slightly more complex in that, in the Volsungs, tasting the dragon's blood grants Sigurd magical powers. In this context, magical powers from the dragon transferred are to Sigurd. This still falls under the umbrella of Sympathetic Magic, but with a different meaning/expression. The interpretation may have something to do with the lack of overt Christian influence in the Völsunga (compare to Beowulf) where the emphasis is on the power imbued in the blood, without moral judgement.
Offhand, I can't think of Chinese tales of dragon flesh being consumed, although there is a famous folktale of a dragon trapped in carp form, who is almost eaten--a perceptive scholar instead frees the magical carp and receives a boon. However, the eating the flesh of the Tang Priest is a central device in Journey to the West. Again, here the magical flesh confers immorality in connection to the enlightened being incarnated in the priest.
DRAGON IN CHINESE CUISINE
As Semaphor so eruditely pointed out, there is at least one Chinese story of actual dragon meat being consumed. [See comments]. There is a tradition in Chinese cuisine of foods conferring blessings, still quite popular in contemporary folk belief. I came across this list of lucky New Year's foods: Noodles=happiness&longevity; Dumplings=wealth; Tangyuan=family togetherness; Niangao: income/status;
Oranges=fullness&wealth; Fish=prosperity. Fish is the flesh typically used for dragon, although lobster, prawns, abalone, etc. may be used, and the association with prosperity almost certainly relates to dragons as rain-bringers in an agricultural society. Dragon & Phoenix still considered a special dish, not just because it's tasty, but because of the positive mythical connotations, and the good fortune they confer. (For those interested, there is a great scene in Eat Drink Man Woman involving "Joy Luck Dragon Phoenix")