Typically I don't have anything that turns me off but it's hard for something to hook me up. I haven't read a lot of stories, but the latest story I tried was Supreme Magus. Read up to chapter 10 to give it a fair chance but it became harder and harder after chapter 3, so here's what turned me off from that specific story. In the prologue we are introduced to the Mc has his 'tragic' life and wakes up in one unfamiliar world where he dies immediately and goes to another. In his first life he is about 50 but his mental age is closer to 14. Ah everyone's my enemy I will kill them all! Kinda vibe. Then no one questions the ability of a three year old to form complete and perfect sentences, his big brother was a giant asshole without the correct backing and the cherry on top would be the inconsistent language of the people. You can't point out how the language is not advanced for Authors inside the world and then proceed to give a random hunter surgical terms as if it were common knowledge.
Sorry for going on and on about something that wasn't asked in the thread. So, to summaries, I guess logical inconsistencies both within and outside of your stories lens, especially early on (because later on you can at least bullshit your way out).