As an example, user @BobaFit recently posted an answer to In the US, what do federal politicians commenting on education policy propose the federal government should do? which I find quite compelling, but which lacks sources for the claims that the author makes. Should the mods remove such answers after giving the author sufficient time to add references to their claims?
No, answers should not be removed just because they lack references. Not all answers here will require references and we can find good answers without them.
If an answer needs references it can be dealt with as needed by requesting them and voting (down and deletion votes) by the userbase without the need for moderators to take action.
Providing links as sources for further information is always appreciated. But I think that not all questions strictly require answers with references. Sometimes a "common knowledge" or "common sense" answer is enough.
If, however, someone in the comments disputes claims made in an answer, then the author should either try to find a trustworthy reference to back up that claim or remove the claim. And if the answer no longer answers the question without the disputed claim, then it should be deleted altogether.
I agree with JoeW's answer that
No, answers should not be removed just because they lack references.
And if one feels that moderator action is required, Stack Exchange already provides several increments that alert readers without removing the content:
Moderators have additional options where we can post notices. The notices are as follows:
citation needed - This post does not cite any references or sources. Please help improve this post by adding citations to reliable sources. Unsourced material may be challenged and removed.
current event - Post is related to a rapidly changing event.
insufficient explanation - We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed.
I've never seen the "insufficient explanation" banner, perhaps it's specific to Unix SE, but I've seen several instances where the other two were used and I think the've been quite effective; they're generally associated with low upvotes and comments specifically asking for lack of sources to be rectified.
An unsourced answer with a "does not cite any references" banner, low votes and comments remaining visible also offers a chance for other readers to remember that this is a problem. SE does not require anyone to know all the ins-and-outs of SE before starting to write posts, we basically learn on-the-fly and these little tidbits have educational value.