You ask "Can you give me any more examples" when you want extra or additional examples BUT you are not sure the person can do it for some reason. This is a more polite and less assertive request, because you are indicating that you are aware that the person may not be able to do it. You ask "Can you give me more examples" when you simply want extra or additional examples. Unlike the previous sentence with 'any more', you have no idea or don't care if the person can provide them. This is a slightly less polite request, as you are asking for it without asking if it is possible to do it. Because of this, it is slightly a more assertive request. "Do you have anymore apples ?" I am asking if they have extra apples but I think there's a chance they don't have any extra apples or are unable to provide them. Asking them if they are able to do it, makes the request a little more polite. "Do you have more apples ? I am only asking if they have extra apples. I have no idea if they do or they don't. I'm at a party and I ask the host, "Do you have any more beer ?" I want some more beer, but I'm not sure if there is any left. Using 'any more' is a little less assertive than 'more'. I am saying I want more beer but I know there's a chance it's not possible for some reason. I'm at a party and I ask the host, "Do you have more beer ?" Here I'm simply asking for more beer. It is a little less polite. If you said it, you would have to be a little careful with intonation so you wouldn't be misunderstood as being rude.

They are similar in this case. The word “any” really isn’t needed, but it’s common. If you stress the word “more” in either “more” or “any more” you’re acknowledging that there were examples given but you’d like even more if possible. If you stress the “any,” you’re saying that even one example more could be quite helpful.