We can probably use a better word than "annoy." Old Major was challenging and warning the animals on the farm on account of their situation. He wanted them to see that their situation was not a good one. His teachings basically had three points.
First, he says that the animals only work without any taste of their labor. And what makes it worse is that they die when it is all over. He says:
‘Now, comrades, what is the nature of this life of ours? Let us face it: our lives are miserable, laborious, and short. We are born, we are given just so much food as will keep the breath in our bodies, and those of us who are capable of it are forced to work to the last atom of our strength; and the very instant that our usefulness has come to an end we are slaughtered with hideous cruelty.
Second, he says that the main problem with their lot in life is man. Humans are cruel masters. They enslave the animals and use them. In fact, man is the only one that consumes without producing. In short, Old Major attributes everything evil to man.
Finally, he calls the animals to action. If they get rid of men, they will be happy. However, the animals must be careful, not to become like men.
Based on these points, some might say that Old Major was annoying them, but others would say that he was challenging them to take a step in the right direction for their own freedom and well being.