The major issue here is precisely what is considered Byzantine and what is considered Roman. If one dates "Byzantine" to Constantine, one could point to the Christianization of the Roman Empire as "Byzantine. If one chooses 476 and the Fall of Rome as the date in which the Byzantine Empire became an entity independent of a larger "Roman Empire," the Theodosian law code would be Roman. The Justianian Law Code is considered a major Byzantine achievement, though less for its originality than for its comprehensiveness as a compilation of existing laws.

Several important theological points were established in the seven ecumenical councils, convened under the Eastern Roman/Byzantine Emperors, including the Nicene and Athanasian Creeds. Caesaropapism was another significant Byzantine innovation.

In literature, the Byzantines collected and preserved many early Greek texts and were known for their literary and textual criticism more than for original literary works. Byzantium produced several notable historians including Michael Psellos, Theophanes the Confessor, Procopius, Michael Attaleiates, and the princess Anna Komnena.

This is a great question. I would say that the three greatest intellectual achievements of the Byzantine empire are as follows.

First, the Byzantine empire gave us law codes. For instance, one of the most influential books ever written was Justinian's Digest. It codified Roman law, opinion, and precedents. This was a massive intellectual achievement that shaped Western culture. The importance of this work cannot be overstated.

Second, the Byzantine empire preserved much of Greek and Latin literature. They did not "fall" like the Western Empire. In fact, they continued in their scholarship and passed it on. And when Constantinople fell, many scholars moved to the West and brought their learning.

Third, the Byzantine empire passed on a great theological tradition from Greek speaking church fathers. In this way, they have developed an important theological voice.

What are some Byzantine achievements?

This is a great question. 

First, Byzantium preserved the culture of the Roman empire (Roman and Greek learning). When the Roman Empire in the West collapsed in 476, Byzantium ensured transmission for future generations. This point alone is one of the greatest achievements of Byzantium. In connection to this, the Roman law code was transmitted. Emperor Justinian's work on the law code was prodigious, to say the least. When we add the fact that many Western countries used these law codes as the basis of their law codes, the significance of this act becomes enhanced.

Second, Byzantium also revived Europe through the Renaissance. When Constantinople was sacked in 1453, many of the intellectuals in Constantinople moved to the continent and they spurred on greater learning. Without this impulse toward intellectual fervor and creativity, the Renaissance may not have happened. 

Unpacking the implication of these two points will yield great insights.