In The Alexiad, Anna Komnena's 12th century history of the reign of her father, Emperor Alexios I, Anna Komnena quotes Homer's poetry at several points: ancient Greek poetry was still being read by educated Greeks at the time. Niketas Choniates, a Byzantine historian who lived a little bit later (1155 to 1216) also references Homer other ancient Greek works of literature in his history of the Byzantine Empire.

Greek science is a bit more confusing. Like you mentioned, most of the documents we have on ancient Greek science don't come from Byzantine-era reproductions, but from the House of Wisdom, which was the center of learning in Baghdad set up by the Abbasid caliphate. The Arabs had no interest in ancient Greek poetry, but they were interested in science, philosophy and mathematics, and in the House of Wisdom many ancient Greek scientific/philosophical works were translated into Arabic. When ancient Greek scientific works were rediscovered in the west during the renaissance it was through re-translations of these Arabic versions. There were no Arabic translations of Greek theater and poetry though, so those came directly through Byzantium.

I don't know when and why Greeks stopped preserving ancient Greek science. When the Abbasids translated various Greek scientific works into Arabic they did it through Nestorian and Syriac Christians, who would have been former subjects of the Byzantines (or would have had some cultural exchange with them) and thus able to read Greek and have access to Greek books. This implies that at least during the 8th and 9th centuries Greek scientific works were still being preserved in the Byzantine world, but it is hard to say when these documents were lost to the Greeks and only preserved by the Arabs.

If anyone knows more about how ancient Greek science was lost in the west I'd be interested in learning about it.

The Byzantines definitely enjoyed the classical stuff they could read in Greek. The Venetus A, one of the best premodern editions of Homer (i.e., the text of the poem, plus glosses and commentary from four ancient sources) came from Byzantium - this kind of manuscript would require a huge amount of effort to compile and copy, and it was meant to be consulted. I'm pretty sure Anna Komnene, as /u/MurdochAV mentioned, cites Thucydides and Herodotus in addition to very frequent Homeric references. There was a widespread attempt to reconcile ancient thought with contemporary/Christian mores. Nonnus of Panopolis rewrote the Gospel of John in dactylic hexameters, and I've heard of (but never actually seen a citation for) something like cento miracle plays, verses of ancient dramas rearranged to tell e.g. the story of the Nativity, or things like that. Platonism was very popular up until at least the 11th century, and it pops up everywhere in Byzantine theology, from justifying the veneration of icons (the theory of Forms) to explaining the various angelic orders in Heaven. Of course there were always a few hard-line ecclesiastics who opposed ancient pagan stuff, but they never managed to stamp it out completely.

I just don't know much about science, modern or ancient, so I won't offer any potentially wrong information, haha. If you're interested in an in-depth modern examination of Byzantine reception of ancient culture, though, I recommend Anthony Kaldellis' Hellenism in Byzantium: The Transformations of Greek Identity and the Reception of the Classical Tradition, which actually addresses this exact question.

Many thanks to MurdochAV and arivederlestelle for their very informative and helpful answers.