Earlier last week, in a Google Hangout on Google+ at the 23:30 mark, John Mueller said Google does recommend that when you move a domain name, that you try to get those linking to the old domain to link to the new one. But if you do not, John Mueller said if you don't you wouldn't "see any critical loss by not doing this."
Here is the question:
How much should I care about updating incoming links after changing domains? A website that have natural links from lot of domains changed domain. All links are redirected with 301. Should we expect any search results visibility boost after updating links?
Here is his answer:
In general, this is something that just makes it easier for users to actually find your content. And if, over time, you're kind of collecting this cruft of redirect chains from one domain to another, then that's something that's probably worth cleaning up at some point.So we do recommend updating the links from external sites to point to your new domain. That does help us to kind of reinforce that this is a permanent move, and you're really moving to this domain, and we should really switch our URLs that we have indexed from the old version to the new version.
So all of these things kind of add up. I don't think you'd see any critical loss by not doing this. But I'd really kind of look into your analytics and see where most of the users are coming from and make sure that they don't have to be bounced through this additional redirect, which means another DNS lookup and all of that.
Here is the video embed:
Forum discussion at Google+.