Q: If Ramadan comes and someone still has days of Sawm (Fasting) to make up for from the previous Ramadan, arethey considered sinful for not having made up the days before the start of the next Ramadan? Do they have to make a Kaffarah (expiation) or not?
A: Everyone who has days to make up from previous Ramadan has to make up for those days before the next Ramadan. They may delay making up for them until Sha‘ban, but if the next Ramadan comes and they still have not made up for those days, without an excuse, they are considered sinful for that. They should make up for these days later and also feed a needy person for each missed day. This was the Fatwa (legal opinion) given by a group of the Sahabah (Companions of the Prophet). The required amount of food is half a Sa‘ (1 Sa‘ = 2.172 kg) of the staple food of your country to be given to some or just one needy person for every day not fasted. If you are excused for the delay by reason of illness or travel, you just have to make up for the Sawm; you do not have to feed needy people. This is according to the general rule in the Saying of Allah (Glorified be He): “.. and whoever is ill or on a journey, the same number [of days which one did not observe Saum (fasts) must be made up] from other days.” (Soorah al-Baqarah, 2:185) Allah is the Grantor of success.
Source: Fatwas of Ibn Baz, Al-Da’wah, vol. 2, pp. 158-159