Near hates L because he is the reason that Mello hates him.
He hates how similar they are. Quiet, intense, and everything else. They were good at concealing their emotions (Near is better, while L is better at reading them in others). They both apparently also have a weakness for their opposites (Near for Mello, L for Kira, Near is sure of it, it is the only reason for the case taking L so long). They were both abandoned as infants, knew nothing of their families.
He hates how L always remembers his birthday, and that birthday and Christmas gifts from him are always the best, the only puzzles that ever challenge him. They are always handmade and always different every year. Unlike himself, Near knows that L is kind, and he hates that as well.
He hates how whenever he needs to know something, L is the one he has to go to, because if Near himself doesn't have the answer, it is a slim chance that anyone else will.
He hates that L has left this place and is free. Because he is so similar, Near can see that L also hated this place, still hates it, and now he is free and an example to them all and Near hates him for it.
He hates that Mello likes L. For Mello's birthday, L sends expensive chocolate, or something to do with rock music, or fashion. Mello is always delighted. Near knows nothing about these things. He hates that L also likes sweet things. It is something he shares with Mello, and, aside from some particular chocolate, Near never had a sweet tooth.
He hates that L is the only one to have seen him cry, really cry, when he hadn't been at Wammy's for very long and he was lonely and Mello hated him, and he broke and sobbed into L's neck and told him everything. And the worst part was that L understood.
Near hates how he, they, are all raised for the moment that L dies, because he knows that L knows this and that it makes him sad, although he never shows it. Near always stokes this up and plans to say it to L's face – stupid L – and rub his nose in it, to see what he will do, but even with all this hate, knows he never will.