How did Byzantine Greeks regard ancient Greek civilization?

As a complement to Dimitra Triantafyllidou’s answer and Niko Vasileas’ answer:

There was an undercurrent of resentment of the ancients and their pagan wisdom, but it remained an undercurrent.

There’s the renowned hymn on the Pentecost by Romanos the Melodist, dismissing ancient learning with puns on the pagan scholars—and alas, a favourite of the Greek nationalist blogosphere:

Οὐκοῦν εδόθη αὐτοῖς πάντων περιγενέσθαι
δι’ ὧν λαλοῦσι γλωσσῶν;
καὶ τί φιλονεικοῦσιν οἱ ἔξω ληροῦντες;
τί φυσῶσι καὶ βαμβεύουσιν οἱ Ἕλληνες;
Τί φαντάζονται πρὸς Ἄρατον τὸν τρισκατάρατον;
Τί πλανῶνται πρὸς Πλάτωνα;
Τί Δημοσθένην στέργουσι τὸν ἀσθενῆ;
Τί μὴ ὁρῶσιν Ὅμηρον ὄνειρον ἀργόν;
Τί Πυθαγόραν θρυλλοῦσι τὸν δικαίως φιμωθέντα;
Τί δὲ καὶ μὴ τρέχουσι και σέβουσιν οἷς ἐνεφανίσθη
τὸ Πανάγιον Πνεῦμα; (On the Pentecost XVII)

Was it not granted to them [the apostles] to be superior, through the languages they spoke in? And what are the fools outside arguing about? What are the Hellenes [Pagans] bloviating and blabbing about? Why does their fancy go to Aratus the accursed [triskataraton]? Why are they deceived [planōntai] to follow Plato? Why do they care about Demosthenes the weakling [asthenē]? Why don’t they see that Homer is an idle dream [oneiron]? Why do they keep going on about Pythagoras, who was justly muzzled? Why won’t they run and pay respect to those to whom the Most Holy Spirit appeared?”

The fact that Romanos was Syrian is not relevant; so was Lucian. The fact that Romanos was writing in the 7th century is relevant: there were still pagans in the Empire, and Christianity was still trying to assert itself.

This was not the elite response to antiquity: the elite response, as Dimitra said, was to embrace antiquity, and the Cappadocian Fathers, Basil of Caesarea and Gregory of Nazianzus, pioneered the reconciliation of Christian and Pagan learning in the 4th century. But Romanos was not part of the elite.

I’ve elsewhere spoken of how Modern Greek peasants were in distant if suspicious awe of the ruins around them: Nick Nicholas’ answer to How do Greeks feel about references to Ancient Greece?

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…

The Parastaseis syntomoi chronikai, a 9th century description of the sights of Constantinople, shows a similar confused apprehension of the highlights of the Ancient World that Constantinople was strewn with: little-understood receptacles of magic and fear. Like Romans, the commoners of Constantinople were ambivalent about their past.

And of course, there was the ongoing feeling of inferiority towards the ancients, memorably expressed by Theodore Metochites: Nick Nicholas’ answer to Do many modern Greeks feel a sense of failure or perhaps inferiority when compared with their ancient Greek ancestors? The ancients have not left us anything to say, he laments—in the introduction to an 800-page collection of essays.