As in Ancient Rome, slaves in Anglo-Saxon England could gain their freedom:
The manumission (freeing) of slaves was solemnized by ceremony, the presence of witnesses, and legal documentation...
The freeing ceremony could be performed in church, or at a cross-roads – symbolic that the freed now chooses his or her own path.
slaves who had gained their freedom would become part of an underclass of freedmen below the rank of ceorl.
Note: ceorl or churl = ordinary freeman, "a non-servile peasant",
According to the article Roman freedmen – slavery in ancient Rome, the book The Freedman in Roman Art and Art History (pdf), as well as Freedom and Slavery in Roman Law and Oxford Classical Dictionary, Roman freedmen sometimes (but not always) had some legal and / or social obligations to or dependency on their former owners as there were different conditions of manumission. These might include having to work part-time for the former master, restrictions on who a freedman could marry and legal provisions concerning property. It was also common for them to take their former master's name (e.g. Marcus Tullius Tiro was freed by Marcus Tullius Cicero).
Did freedmen have any obligations to their former masters in Anglo-Saxon England? If so, do we know what they were?
For the time frame, any time from the rise of the Anglo-Saxon kingdoms in England to the Norman conquest would be of interest, though I doubt much is known about the early period. I'll narrow this if it proves too broad (which may be the case if the Viking presence changed things).
Note: I did not make it clear in the original wording of this question that I am citing Rome merely as a source of examples of what obligations freedmen might have. In clarifying this point, I do not in any way intend to invalidate Pieter Geerkens' useful and informative answer.