1: The destruction of the relays basically kills EVERYTHING around them, I believe it was in the Arrival DLC that by destroying the relay in that system its explosion kills off most of the Batarian race and homeworld (this is gathered from reading). This is because the relays themselves contain so much energy that a detonation basically spells doom to the galaxy it is in.
2: The reapers were supposed to be about stopping the flow of Dark Energy, something alluded to in Mass Effect 1 and 2, but Mass Effect 3 just kind of stops talking about it. The dropped plan for the ending though does address this fact and allows you to make choices on how to deal with the Dark Energy problem (aka, kill reapers and let the universe either fight off Dark Energy or die trying/ Let the reapers continue what they are doing so as not to snuff out every living through and destroy the universe/ etc etc) Mass Effect 3 is a violation of Chekhov's gun.
3:The premise that God Child speaks to you about, the overwhelming need for synthetics to destroy all organics makes no sense and is very convoluted in a lot of sense (insert Xzibit picture here) not to mention that Mass Effect 2 and 3 basically show you that it is indeed possible for synthetic life to coexist with organics and not every cycle is doomed.
4:Concerning the synthesis ending, it goes against everything the series has been based around, which is that through diplomacy and choice, all races can come together to coexist without having to conform to a single standard...instead this ending is basically Shepard saying "fuck everyone, I make the choices, and negating any freewill of the entire universe"
5: Joker fleeing the battle, this is a major one people touch on..especially me as the scene showed the 2 squadmates I had on earth getting out of the ship with him. It's not feasible that he flew down into London, picked up those 2 members then booked it out of the Sol system leaving the fleet to die against the reaper horde.
6: Thannix...missles? In Mass Effect 2 you have a Thannix cannon, which is basically equiv to the reaper laser beam of doooooom, but it was retconned in mass effect 3, not really a plothole but more of a ..."wuh?"