The other day it was brought up in EC chat that shall is used in asking permission. At that moment I thought about it for a minute and said I didn't think shall was ever used in a question for permission. This lead me to look into it further and I found my thinking was correct. I could find no examples of shall being used in this way. So where did they get this idea from? It was from the construction such as: Shall I open a window?.
At first glance I can see how this is taken as asking someone for permission, and by itself it does kind of seem that way, but actually you aren't really asking for permission. It's close, but it has a different use. Shall I in a question can do 2 things; It can make a suggestion or be an offer.
Take the following example. It's a bitterly cold winters day. Two teachers, Lisa and Helen, are making their way to a room to get it ready for a class function the next day. It has those old hateful radiators that usually make the room way too hot. When they enter the room the conversation goes like this.
Lisa "Wow, it's really hot in here."
Helen "Yes, someone has left the radiators on full blast
again."
Lisa goes to turn them off and says "It's going to take 30
minutes for them to cool down."
Helen "Shall I open some windows?"
Lisa "Yes,please do."

Actually in this case Helen is not asking for permission, she is merely offering a quick solution that will help the situation. Helen already knows Lisa probably wants those windows open, she is just offering to help as Lisa goes around turning off radiators.

If you put it in another context it becomes more apparent it is not asking permission, or at least to me it is apparent.

Some students are in the lobby of a University sitting at a table. They are half way studying for exams and chatting about other things going on in their lives, as students often do. Then John says something like this:

John: "OH no, I forgot a pen." He turns to Jeff sitting next to him and says "Shall I barrow your pen for a second?"

This sounds really strange, and Jeff would probably look at him with a funny expression on his face. It sounds like there are some known alternatives that could help with John's situation, and he is asking Jeff if borrowing his pen would be a good option to choose. It also could sound like John is asking Jeff if it's a good idea that he borrow his pen, as opposed to taking some other action.

So here are some examples of how this works.

Shall I order a taxi? offer
Shall I help you with your suit cases? offer
Shall we begin the meeting now? suggestion

Someone said "Well maybe it's just different in British English?" I could find only one difference concerning this situation. In the United States we tend to shy away from shall and use should instead. However, sometimes in the U.S. you will hear shall used for I or We, almost never for: you, he, she or they.

So in the U.S. it's:

Shall/Should I call her on her cell phone?
Should you call her on her cell phone?
Should he call her on her cell phone?
Shall/Should we call her on her cell phone?
Should they call her on her cell phone?

In British English it seems they mostly use shall in all cases above. However, whether it's said "Shall he call her cell phone?" or "Should he call her cell phone?", it's the same thing and is a suggestion(like in the previous questions), or is an offer.

I really tried to find shall used for permission but I just couldn't find any examples. If I'm missing something let me know.