while this would not immediately ruin the byzantines, i believe it would in the long run:
the visigoths settle in the balkans and do not sack rome, and do not move into gaul and spain. they might attempt a siege on constantinople.
the ostrogoths might do likewise.
attilla the hun lays siege to constantinople. he fails, and turns around to ravage greece.
gepids, lombards, sciri, alans, sarmatians, and huns are all other possible groups who may decide to settle in the balkans instead of western europe because of the wealth there.
later on, the magyars, avars, and maybe even mongols could do the same.
Doubt it. The Byzantines did perfectly well without effective control over the Balkans in the ninth and tenth centuries: in the great scheme of things, Greece and the wider Balkan peninsula were near worthless compared to the real heartlands- Thrace, the Aegean Isles, and Anatolia. To knock out Byzantium, it is these regions that really have to go.