What happens to the eggs after a woman has a tubal ligation?

Fallopian tube. Ovulation occurs into the pelvis (peritoneal cavity) and ovuum pickup is handled by the fimbriated end of the fallopian tube. After succesful ligation the tube is blocked in the mid-segment so after pickup sperm and egg cannot meet. An ovuum that is unfertilized within possibly as short as 10 hours dies and is resorbed.
Eggs are recycled. An ovulated egg that does not find sperm simply melts away. The egg has a limited life, and if not fertilized, it is recycled. Nothing unique about the tubal ligation. This happens with every egg that is not fertilized every month.