Why does Antigone feel it is her duty to bury Polynices?
Most obviously, Antigone feels duty-bound to bury Polynices because he is her brother, as the previous answer states. This brings her into immediate conflict with the city's ruler Creon who decrees that the corpse of Polynices should be left to rot outside the city gates because he was a traitor to the city. In fact, he died fighting Antigone's other brother Eteocles who was honoured for defending the city. Antigone has already buried her parents and Eteocles, and now she wishes to do the same for Polynices, despite Creon's edict. She willingly takes on Creon and refuses absolutely to back down.
Antigone at times also intimates that she is so determined to bury Polynices because it is an offence against the gods, against morality, to leave any corpse unburied. 'Death longs for the same rites for all,' she informs Creon. Actually, she does appear much obsessed by death; she invokes the gods and the dead many times. She seems to have more of a connection with the dead than with the living. This no doubt has much to do with the fact that her own family is so doomed - her father Oedipus committed incest with her mother, who committed suicide when finding out the truth, her two brothers killed each other, and so on. She is obsessed by the peculiarly grim fate of her family and seems to be lured down a similar path.
We should also remember that mourning the dead, performing funeral rites, was the province of the women in ancient Greece (and in many other cultures, both in the past and present). Antigone, as a woman, therefore would feel a greater obligation to perform the last rites for her brother, and she is scornful of her sister Ismene for refusing to do the same.
Interestingly, when being led off to her entombment, Antigone observes that she never would have defied Creon to bury a husband or children, only a brother - her reasoning being, apparently, that if her husband or children died, she could get another husband, or bear more children to another man, but, because her parents are both dead, it's impossible for her to get another brother. This admission underlines the feeling that she is so determined to bury Polynices not in the interests of upholding morality in general but because she really cares about her own family alone, and particularly about her own dead.
There are several reasons why Antigone feels that it is her responsibility to bury Polynices.
First, we need to remember that Polynices is Antigone's brother. So, she feels a strong attachment to him. There is a blood relation that cannot be minimized. So, we can say that there is filial duty. Antigones must be a faithful sister.
Second, according to the Greek world, one of the worst thing that you can do is not bury a person. It was believed that that unburied person would not find rest. This is why it was so important that Priam get his son's body (Hector) back from Achilles. Similarly, to Polynices to rest properly, there needed to be a burial.
In view of these two reasons, Antigone seeks to bury the body of Polynices no matter what. She shows great courage. As for her sister, Ismene, she is cowardly.