From the spec
10.3.2 301 Moved Permanently
The requested resource has been assigned a new permanent URI and any
future references to this resource SHOULD use one of the returned
URIs. Clients with link editing capabilities ought to automatically
re-link references to the Request-URI to one or more of the new
references returned by the server, where possible. This response is
cacheable unless indicated otherwise.
So yes, if you are permanently redirecting a URL then you should update links to point to the new location.
In practical terms, this will save browsers visiting the link having to ask for the document only be to told to go away and ask for it somewhere else repeatedly in a digital kafkaesque nightmare.
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solved What is the best way to do this site redirection