Neither of the twins like / likes apples?

It can both, I think. But being the eldest of twins myself, I would go for likes because it kind of more expresses that 'neither he and she' likes apples (two persons who don't like apples), while 'neither do they' like apples kind of gives the impression twins are one instead of two. To clarify it a bit, since I was 12 I have always wanted to be seen as a single person without my twin sister, and I don't really doubt that other twins want that as well. So that is why I would go for 'likes'.

Hope this was a helpful answer ;-). I'm not really good at the spelling or grammar of words, but I think this was okay.