I wrote quite a bit about 'Byzantine' consuls here and their Senate here. In both cases they were certainly much less powerful than their Republican predecessors, but they were nevertheless rich and influential men, so they all continued to play a role in politics, just like aristocrats everywhere do. The prestige of the consulship would be why Justin II revived the office and made it an imperial prerogative, as well as why Heraclius made sure his rebellion's propaganda was filled with consular imagery. Image matters both now and then, so it is no surprise that a powerless office was still seen as an important one. The same goes for the Senate and, as I argued above, it did play a significant role in the sixth and seventh centuries, particularly when the emperor was weak. One thing that I would change about my answers is everything I said about their status after the seventh century, as they are outside of my field of interest and I definitely haven't done enough reading on them.

The senate was formally stripped of its official power by Leo VI in the "basilika" law code in the late 9th century. From that point on being a senator gave you no de jure power in politics. That doesn't mean that the senate wasn't influential though, since the majority of senators were very wealthy and/or held a position in the higher bureaucracy.

The senate as an institution independent from the rest of the administration had effectively ceased, and the position of "senator" was a purchasable title instead of a job. References to the "senate" doing something in middle Byzantine literature are, more often than not, better described as the officials of Constantinople doing something. This was the case in the fourth crusade, where the magistrates stationed in the city assembled and decided on who the new Emperor would be.