I think the important thing to look at is the context of Franco-Byzantine relations in the 8th and early 9th centuries. Throughout the period, from 754 onwards the Byzantines and Franks had been vying for control over Italy. In 756 after the Franks, led by Pippin III, had defeated the Lombard kingdom messengers from the Byzantines appeared, asking them to hand over the lands that had formerly made up the Byzantine Exarchate (which had been conquered in 751 by the Lombards). Pippin III did not hand over the lands. Later in the 8th century the Byzantines would attempt to interfere in the politics of Italy once the Franks had formally annexed the Lombard kingdom in 774/6. In 788 they sent an army led by the son of the deposed king of the Lombards. In short, things were not rosy between the two even before the coronation. There is certainly more to be said about the relationship between Byzantium and Francia but that will suffice for now.
As for the coronation itself there has been a lot of debate about why it happened and whether Charlemagne knew. For the most part historians now agree that Einhard was probably not telling the truth when he said Charlemagne had no idea he was to be crowned emperor. The question of why is a bit more difficult to answer, and that comes down to the sources being problematic. The Annals of Lorsch would appear to give us a clear cut reason: the eastern empire was ruled by a woman, Irene, and thus needed an emperor. However, Janet Nelson and Roger Collins have both written about the danger in attempting to use the AL. Nelson and Collins both followed a similar approach, that the AL may not be entirely accurate because the different sources we have likely represent different interpretations and narratives of the event. However, that still does not tell us much about the actual reaction of the Byzantines.
Constantine Tsirpanlis wrote an entire article just devoted to the topic (handily titled "Byzantine Reactions to the Coronation of Charlemagne (780-813)"), where he argued that the Byzantines saw the coronation as a threat to the eastern empire. I tend to agree with Tsirpanlis, it would appear that after the coronation there was a lot of uneasiness amongst the eastern empire. They send embassies in 802 and 803 in order to make peace with the Franks, although the immediate context of this treaty has been lost. Tsirpanlis also links this to the coronation. After the coronation it would appear that the Byzantines took a much more defensive stance towards Francia, not only in more aggressively sending fleets into the Adriatic, but also in reorganizing some of the themes in order to be better defensively organized.^1 Finally, Einhard would seem to preserve a rumor that was going around at the time that Charlemagne was planning to invade the Byzantine empire. He writes that when Charlemagne became emperor "he aroused their [the Byzantines] strong suspicion, for he might well have been planning to take their own imperial power from them." Tsirpanlis, focusing more on the eastern perspective, would agree that this was a very real worry for the Byzantines in the immediate aftermath of the coronation, and fears of a reunited empire between Irene and Charlemagne may have led to Irene's overthrow.^2
Indeed between 800 and Charlemagne's death in 814 there were several conflicts between the two empires. In 806 Charlemagne intervened in Venice and Dalmatia, which were under Byzantine control. Promptly the Byzantines sent a fleet to retake control, and there would be fleets in 807, 809 and 810 before negotiations finally began over the region. It was not until 812/3 that a settlement was reached. The Annales regni francorum preserve some of the provisions in the treaty, although its text has been lost. Apparently, Charlemagne agreed to leave the regions alone and give up his claims there, in return the Byzantines were to recognize his title as emperor. After 812 there are no more conflicts between the Byzantines and Franks until well after his death, and in 839 envoys would show up at the court of Louis the Pious seeking to renew a treaty of friendship. What does this all mean? The simple fact that the treaty led to the Byzantines acknowledging Charlemagne's title indicates that from 800-812 they had not viewed him as a legitimate emperor. Even after there is some debate about whether they fully accepted it, or if they played down his title.
TL;DR The Byzantines were not very happy about the coronation and felt threatened that Charlemagne would sail east to conquer the eastern empire. By 812 the Byzantines were ready to acknowledge his title (at least provisionally) and that is when open hostilities between the two would appear to stop.
Most of the primary sources are the Annales regni francorum, Einhard's Vita Caroli, and the Annals of Lorsch. There is a quite lengthy literature on the coronation and if people want some further reading I can certainly provide more to read.
*1) W. Treadgold, The Byzantine Revival, pp. 159-61. *2) Tsirpanlis, "The Byzantine Reactions", p. 69.