Holographic communication was first installed on the USS Defiant during the series Star Trek: Deep Space Nine, which takes place over 100 years after ST:Discovery. However it is commonly used in this episode.
Burnham's radiation exposure injuries are treated with an extended session of antiproton exposure. Per The Deadly Years (1967) the standard medical treatment for radiation exposure had been regular injections of an antiradiation medication called hyronalin for years.
In Kirk's time as captain, each starship had different chest insignia on their uniforms. It wasn't until after Enterprise's refit (2271) that they became uniform (that of the Enterprise). Yet, the Discovery crew are wearing the same chest insignia as the Enterprise 20 years earlier.
The crew of the Shenzhou sees Klingon cloaking technology in action, yet in Balance of Terror (1966), set 10 years later, Starfleet is unfamiliar with cloaking employed by the Romulans.
There is no way the USS Shenzhou could have seen the Starfleet insignia that Burnham and Georgiou's footprints made in the sand from orbit. Massive manmade buildings cannot be seen from space, and there were clouds covering the area (plus heavy winds that likely would have obscured the pattern they'd walked as they were making it).
Despite a claim that Michael Burnham cannot be a mutineer, since the rest of the USS Shenzhou crew never disobeyed the captain's orders, Michael did knowingly attack the captain and tried to take over the ship, which is the definition of a mutiny, so Michael is in fact a mutineer, even if the rest of the crew is not.
Shortly after the Beacon lights up Sarek mentions "the new star", despite being many light years away. Similarly, the Klingons come to the call of the Beacon's very promptly from all the corners of the Klingon Empire. Normally, it would take light years, if not decades, to reach all the parties. Evidently, the Beacon doesn't just light up, but transmits the light via sub-space or using some other faster-than-light method.
In the desert scene first officer Burnham does not realize she and Captain Georgiou have been walking in a "circle" Since Burnham was was raised on Vulcan and attended the Vulcan Science Academy and can discern how long it will take for a sand storm to reach them (to the second) Surely she would have realized they were making very sharp turns at times while tracing out the pattern of the large Star Trek insignia in the desert. They would have had to have made several very sharp course changes at the points of the insignia. But it is reasonable to assume that Burnham is using the phrase "walking in a circle" colloquially to simply mean, "we've returned to our starting point," the same way everyone else uses it.
Lieutenant Commander Saru is addressed as a lieutenant on one occasion (09:23). This happens frequently in other Star Trek series as well. Apparently, in Starfleet it is acceptable to refer to a Lt. Cmd. either as a Lieutenant or a Commander.
The tagline of Star Trek Discovery states "Ten years before Kirk, Spock and the Enterprise.." in the established continuity of Star Trek the Enterprise was already in service with Mr Spock as science officer at the time Discovery is set. Robert April was the first Captain of the Enterprise NCC-1701 which Launched in 2245 , If it is set in 2255 then the Enterprise is already in Service with Robert April or Christopher Pike in command. But clearly the tagline means before Kirk and Spock are together on the Enterprise.
General Order One, aka the Prime Directive, forbids interference in the development of pre-warp civilizations, regardless of intent. Burnham and Georgiou blasting open a well on an undeveloped world violates that directive.
The Klingons load the corpse into a coffin to be anchored to the exterior hull of the ship. Per Heart of Glory (1988) and Emanations (1995) Klingons do nothing to their dead, believing the corpses to be nothing but empty shells despite a belief in an afterlife.
In T'Kuvma's opening speech, the closeup of his face shows his nose/lower brow ridge to be moving as he speaks as though the prosthetic is trying to come off.