Would ouyou mind getting me a newspaper?
Will you mind getting me a newspaper?
Both the phrases "would you mind" and "will you mind" are grammatically correct. It's up to you whether to choose the phrase that's more common and idiomatic (the former)) or
The use of the phrase that's unusual or far less idiomatic (the latter) when"woud you request someone politely to do somethingmind" is more usual as it's more polite.