Most of time, 请 is used for politely demand/command nowadays. E.g. A teacher can say 请不要说话了. This demand is much milder than 闭嘴！
请把我的手机拿过来 can be a milder way to order or command. The tone plays an important role as well.
in your circumstances, I would probably say:
麻烦帮我把手机递给我. 麻烦 connotes to trouble you(sorry to bother)，帮我: to help me(not your ...