Some of the Fathers believed in flat earth, but the flat earth proponents exaggerate how common it was. For example, they misinterpret some passages as being in support of flat earth that are not actually saying that. For another example, in the above post by happenby, she places Cosmas in her list of Fathers, but he is not one.
I think the Wikipedia list of Fathers supporting flat earth is more reliable than flat earther sources.
https://en.wikipedia.org/wiki/Flat_Earth
The more useful writings of the Fathers on the subject put it in its proper perspective. For example, this passage by St. Basil in the Hexameron presents it well:
Those who have written about the nature of the universe have discussed at length the shape of the earth. If it be spherical or cylindrical, if it resemble a disc and is equally rounded in all parts, or if it has the forth of a winnowing basket and is hollow in the middle; all these conjectures have been suggested by cosmographers, each one upsetting that of his predecessor. It will not lead me to give less importance to the creation of the universe, that the servant of God, Moses, is silent as to shapes; he has not said that the earth is a hundred and eighty thousand furlongs in circuмference; he has not measured into what extent of air its shadow projects itself while the sun revolves around it, nor stated how this shadow, casting itself upon the moon, produces eclipses. He has passed over in silence, as useless, all that is unimportant for us. Shall I then prefer foolish wisdom to the oracles of the Holy Spirit? Shall I not rather exalt Him who, not wishing to fill our minds with these vanities, has regulated all the economy of Scripture in view of the edification and the making perfect of our souls? It is this which those seem to me not to have understood, who, giving themselves up to the distorted meaning of allegory, have undertaken to give a majesty of their own invention to Scripture. It is to believe themselves wiser than the Holy Spirit, and to bring forth their own ideas under a pretext of exegesis. Let us hear Scripture as it has been written.
http://www.newadvent.org/fathers/32019.htm
St. Basil cautions against reading meanings into Scripture about the shape of the earth, saying it is a vanity that will not lead to the edification of our souls.
St. Basil was making a point about keeping people on track for what was important at the time. He was not trying to debunk the truth, nor suggest that it didn't matter whether we believe in the earth or globe. After all, suggesting he didn't really care is ridiculous because we all know it really does matter: One is true, the other isn't. It just so happens that St. Basil showed his flat earth favoritism elsewhere.
St Basil speaks here of the firmament in regard to the tabernacle: "In the midst of the covering and veil, where the priests were allowed to enter, was situated the altar of incense, the symbol of the earth placed in the middle of this universe; and from it came the fumes of incense." (The Mystic Meaning of the Tabernacle, Bk V, Ch VI; Clement of Rome, Stromata, Bk V)
St. Basil was a full on flat earther and influenced Cosmas' book Christian Topography 200 years later by providing the knowledge of the typology regarding the Church, the tabernacle and the earth. Thank you St. Basil!