Yes, it became virtually non-existent. Once the Byzantines stopped using the remants of Latin in military and legal terminology, there was little utility in them learning Latin. The late Latin legal texts were translated/summarised into Greek, and there is no surviving textual evidence of a tradition of Byzantines learning Latin (Elearnor Dickey talks about this in her recent book "Learning Latin the Ancient Way", btw.).

Byzantines in the post-antiquity period had little reason to learn Latin and so generally did not.