残り and 余り: the part remaining and the amount left over
Distinguish unfinished work or time remaining from a surplus after distribution, recognizing that the same remaining items may take either word in context.
A portion not yet dealt with or used
残り, read のこり, means the remaining part or amount. 残りの仕事は明日します。 says that the remaining work will be done tomorrow.
残り時間は10分です。 says that ten minutes remain. Distinguish having some time left from judging that it is enough for the required work.
What remains after distribution or use
余り, read あまり, can describe what remains after the required amount has been distributed or used. 十個のパンを七人に一個ずつ配ると、三個余ります。 says that giving one piece each from ten pieces of bread to seven people leaves three.
七を三で割ると、余りは一です。 says that seven divided by three has a remainder of one. 余り is also used for the remainder in division.
A remaining amount and an unfinished task
Bread left after distribution can also be called 残りのパン. Do not insist that 残り and 余り refer to different objects; consider the focus on what remains or what is left over after distribution.
残りの仕事が多いです。 says that much work still remains to be done. It does not say that the work is an unwanted excess beyond what is needed.
Does 残り in 残りの仕事は明日します。 refer to unnecessary work to discard?
Show the answer and why
No. It refers to work not yet completed, which the speaker will do tomorrow.