The Code of Canon Law prevents certain people from marriage:
Canon 1091 §1. In the direct line of consanguinity [blood relationship] marriage is invalid between all ancestors and descendants, both legitimate and natural.
§2. In the collateral line marriage is invalid up to and including the fourth degree [including the relationship aunt-nephew, uncle-niece, first cousins, and siblings].
While the Church has occasionally dispensed consanguinity in the fourth degree (that is, it has occasionally made an allowance for first cousins to marry), a letter of Pope Benedict XIV exists titled Aestas Anni, which states (in paragraph XIII):
Non convenit inter Theologos an matrimonium inter fratrem et sororem, jure naturali, Divino, aut humano prohibeatur. ... neque tamen ullo probato testimonio evincitur, a quopiam Pontifice admissas fuisse preces fratris et sororis matrimonii vinculo invicem ligari petentium.
That is (my translation):
Theologians do not agree whether marriage between brother and sister is prohibited by natural, Divine, or human law. [The difference is that the Church can set aside prohibitions of human law, but not of natural or Divine law.] ... nor is it demonstrated by any tested evidence that the requests of a brother and sister to any Pope at all, seeking to be bound in the bonds of marriage to one another, have been admitted.
No Pope, in other words, has ever allowed a brother and sister to marry.
Thus, the couple's marriage would certainly be considered invalid; and it would have to be nullified (the Church would recognize publicly that there was never a valid marriage). The question is, what to do now?
The siblings may or may not feel comfortable living together; that decision is theirs to make. If they so choose, it would not be a violation of canon law for them to live together as, well, brother and sister.
Although their marriage was invalid, and therefore null, they entered into it believing it to be valid; the Church calls such a relationship a putative marriage:
Canon 1061, section 3. An invalid marriage is called putative if at least one party celebrated it in good faith, until both parties become certain of its nullity.
The children of the union are therefore considered legitimate:
Canon 1137. The children conceived or born of a valid or putative marriage are legitimate.
And certainly the parents still have the obligation to care for them:
Canon 1136. Parents have the most grave duty and the primary right to take care as best they can for the physical, social, cultural, moral, and religious education of their offspring.
Whether the brother and sister chose to chastely live together in an unmarried state, with their children, or separated and made some other arrangement to raise the children, they would be considered unmarried, with legitimate children, and with the option (should they so choose) to marry a new partner in the Church.