It seems like it would depend on the Midrash. Some of them might conflict with Christianity. For example, some Protestant hymns probably conflict with Orthodoxy, like if they mention original sin. On the other hand, some Protestant songs are on rare occasion sung in some Orthodox parishes, including mine. So it seems like if there was a short simple midrash, then maybe it wouldn't conflict.
Regarding Jewish feasts, it seems some of them might be ok like Hanukah because Jesus Christ was in the Temple walking at Hanukah, John's gospel says. Plus, Paskha is a Christian Pesakh, or in English "Passover". Others like the Day of Atonement would seem less likely, because its meaning of a guilt-sacrifice has been fulfillied in Christianity at Paskha. It's my opinion.
But I agree with posters above that the Orthodox church simply doesn't do the midrashes or feasts like Hanukah in practice, although there are exceptions among individuals. I think that they don't matter in Christianity, although it may be nice or interesting to have them.