As far as I know, it was forbidden to sit in the Azara unless you were a king descended from David.
So it seems the Kohanim must have been forced to stand while eating their portions of the Kodshei Kodashim. But this seems to me like a very undignified way of eating food, and an affront to the holiness of the Korbanot.
Tosafot to Yoma 25a discuss whether the kohanim sat down to eat korbanot in the 'azarah. They suggest three approaches:
They would eat standing up in the 'azarah, due to the prohibition on sitting there. Only kodashim kalim (eaten throughout the city) could be eaten sitting down.
The prohibition on sitting in the 'azarah does not apply to things which are part of the temple service, which includes eating korbanot. Thus, they would sit in the 'azarah to eat korbanot.
The word למשחה ('as a perquisite', Bamidbar 18:8) used in reference to eating korbanot teaches that they must be eaten while sitting, notwithstanding the general prohibition on sitting in the 'azarah. Thus, they would sit in the 'azarah to eat korbanot.