Will the grandmother bear a burden of sin for the death of her sister's son during breastfeeding, knowing that the cause of death was the breastfeeding itself?

The default is that breastfeeding does not cause the death of the infant. However, if it is confirmed that the cause of the infant's death was the negligence of the wet nurse, such as overfeeding the infant, leading to milk solidifying in the stomach until death, then this is considered involuntary manslaughter (قتل خطأ).

In this case, the 'Aqilah of the wet nurse (her male agnatic relatives) must pay the blood money (Diyya). If she has no 'Aqilah, it should be paid from the public treasury (Bayt al-Mal). It is waived if the heirs pardon. The wet nurse is also obligated to perform an expiation (Kaffarah), which is freeing a believing slave. If she cannot find one, then she must fast for two consecutive months.

If the wet nurse passes away, the freeing of the slave is from her wealth. If no slave is found, her closest guardian (Wali) is recommended to fast on her behalf, based on the Ahadith: "Whoever dies owing fasts, his guardian should fast on his behalf." The guardian (Wali) is the close relative, whether agnatic, heir, or otherwise. If a stranger fasts with the guardian's permission, it is valid.

In summary: If it is proven that your grandmother's negligence in breastfeeding her sister caused her death, and your grandmother could not find a believing slave to free as expiation, then it is recommended for her to fast two consecutive months on her behalf. If her sister's negligence is not proven, then there is nothing upon her.