I know that it is polite to say "would you mind buying me a newspaper?".
But can we say "will you mind buying me a newspaper?" though I feel it is uncommon.
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We might say
Will you mind buying me (something)?
but we would much more likely use would you.
It would not necessarily be seen as impolite, though it tends to suggest that the speaker assumes that the listener is going to buy them something.
The use of would communicates remoteness and is generally considered polite because it theoretically makes it easier for the listener to respond with something like Sorry, I can't (because ...).
It's possible that some people would sometimes use will instead of would unconsciously or unthinkingly, and there's a good chance a listener may not notice the difference. But I think it would be relatively unusual.
We would more likely use Will you mind (something) when, for example, I know that you will be doing something in the future. If I know that you are planning to go shopping on a busy shopping day, I might ask Will you mind the crowds?
If I know you are going to visit Antarctica, I might ask you Will you mind the cold?
So this is not a request for someone to do something. It is a question about whether they will be bothered when some expected future event occurs.
It has been portrayed in movies that a good looking woman may ask a man at the bar
Will you buy me a drink?
which is different than the stereotyped
Man: Can I buy you a drink?
Woman: I don't know, can you?
but usually "would you" gets used since it is a hypothetical proposal.
Would you 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.
The use of the phrase "woud you mind" is more usual as it's more polite.