How do Greeks feel about references to Ancient Greece?

Depends, as with many of these things.

Yes, there is the reaction you mention. You will occasionally get Greeks (and non-Greeks) reminding you that the Roman Empire kept going for 1000 years after 476, thank you very much—though the relation of Greeks to Byzantium is more complicated than that.

There is the haunting feeling that we’ll never measure up to the ancient Greeks (one that the Byzantines shared).

There’s the reaction against that, with “we’re sick of hearing about the ancient Greeks”. You won’t get much of that aired to non-Greeks, but if you google  Αρχαίοι Ημών Πρόγονοι (“Our Ancient Ancestors” in Katharevousa) and can translate what is being said, much of it makes fun of Greek ancestor-worship.

It’s a profoundly ambivalent relationship. The unlettered peasants 300 years ago had a much more straightforward relationship with the Hellenes: they were this race of pagan giants, the folk who built all them ruins; and they died out because they fell over, and couldn’t get back up…

EDIT: See also Do many modern Greeks feel a sense of failure or perhaps inferiority when compared with their ancient Greek ancestors? (where I say pretty much the same.)