Is one allowed to violate the boundaries and walk beyond techum Shabbat on Shabbat for the mitzvah of Bikur Cholim (visiting the sick)? Is there a responsa or Halacha brought down on this?
You'd be violating techum before getting to bikur cholim.
Yes, circumcision (if on the 8th day proper) overrides Shabbos, but that's the exception; and even then, only the act itself, not its prerequisites. Say the only way to do the circumcision is to forge a new scalpel; the Gemara says that Rabbi Eliezer allowed it, but the halacha doesn't follow him. The same would apply, for instance, if the only way to do the circumcision is to drive there.