What to call it when someone who refuses to eat when offered food, but not because he doesn't want to eat, but because he's being polite

Since being polite in the West would mean that you should accept whatever food you're offered, the exact opposite happens and people end up eating a meal that they don't like out of politeness! Hence, I don't think there's an exact equivalent for this in English.

The closest word I can think of is coy (adj. someone who pretends not to want something).

There's also the literary term accismus, which is a form of irony in which a person feigns a lack of interest in something that he or she actually desires.

One finds usages such as this from Shimmer, by B.J. Robertson, page 25:

Now you eat. I do not stand on formality.

Or this from Faith, by Donald Eaton, page 97:

Eat up. We do not stand on formality here.

As Jalene commented above, 'stand on ceremony' is also possible (e.g. in Just Maagy by Virginia Stringer, page 19.)

The girl is hungry and we have breakfast ready. Let's not stand on ceremony. Come in! Let's eat!

Also it's not wrong to use the word 'shy' in this situation. For example, here's an indirect use from the novel The Cheetah Chase by Karin McQuillan (1995), page 127:

The princess picked out a date with her long red-enamelled fingernails and popped it into her mouth. "Please, eat." I was famished and didn't need a further invitation. Wynn wasn't shy, either.

In Indian English, it's fairly common to say something like "There is no formality, please eat."