Illegitimate children and their descendants are also specifically and permanently barred from succession. If a monarch marries his or her predecessor's illegitimate child, are the couple's children in the line or barred from succession? If a person who is in the line marries a person descended from an illegitimate child of a monarch, are their children in the line?
If a monarch marries his or her predecessor's illegitimate children, I would call into question the closeness of the relationship of the two before worrying about the succession...
But, okay, let's role with this. Say Queen Victoria married one of her FitzClarence cousins (or a child of a Clarence cousin) instead of Albert. The FitzClarence's were the illegitimate children of William IV, and thus barred from the line of succession, thus making Victoria her uncle's heir. Would the children of Victoria and Fitz have been eligible to succeed?
The short, yes. They wouldn't have been in the line of succession through their father, but they would have through their mother. It's the same as if Victoria had married any other individual not in the line of succession; her children gain their place through her rights, not their father's lack o rights.
Now, say there had been a child of Edward VIII who for whatever reason (illegitimacy, being born to Wallis, etc) was barred from the line of succession. If that child then married Elizabeth II then their children would have had succession rights through their mother, just not their father. The whole "no descendants of Edward VIII an Wallis can inherit" thing refers to the idea that they cannot inherit through their relation to Edward.
I wonder if this sort of thing has ever happened in the UK. The fact that Prince William asked for permission to marry despite being descended from several princess who married into foreign families appears to answer the question. Descendants of such princesses do not need to ask for permission, but William did because he was also a male-line descendant of a British monarch.
William asked because while he was descended from individuals who had married into foreign royal families, his claim to the throne comes first from his father (and grandmother), whose claim comes from other individuals who do not descend from people who are not married into foreign royalty (they may have married foreign royals, but they didn't marry into that family; the DoE became a British royal, while Elizabeth II did not become a Greek one).
This goes back to the same kind of argument I made with the illegitimate children. If William's place in the succession was owing to his relationship to the (originally) Greek DoE, then he wouldn't have had to ask for permission to marry because his descent was through foreign royals. But because his place in the succession is owing to his relationship to the British Queen, his descent is through British royals, so he has to ask for permission.
It'd be the same with the Phillips, who are not male-line grandchildren of a monarch. Their claim comes through their British relations, so they have to ask permission.