(First, thanks to Joan for discussing this with me over at the Richard III Society's private Yahoo group.)
On a number of places on the Internet, including both the Richard III Society and the Richard III Foundation websites, it's stated as a fact that Richard wept openly at his queen's funeral. The Richard III Foundation adds the touch that he shut himself up for three days afterward. None of these accounts cite a source for their information.
So did Richard weep at Anne's funeral and shut himself up for three days afterward? Nothing I have found supports either statement. The Crowland Chronicler simply states, "Queen Anne died and was buried at Westminster with honours no less than befitted the burial of a queen." No list of those who attended the funeral is given, and Crowland says nothing of Richard shutting himself away.
Richard himself did not make such a statement. In denying rumors that he had poisoned his queen so that he could marry his niece, Richard said that he was not "willyng or glad of the dethe of his quene but as sorye & in hert as hevye as man myght be." Nowhere in his expression of grief does he mention his attendance at Anne's funeral or three days of private mourning.
As for modern sources, Richard's biographer Paul Murray Kendall, who deeply admired his subject and who would surely would have gotten the maximum pathos out of Richard's weeping at Anne's funeral or isolating himself, doesn't mention him doing either. Joanna Laynesmith in The Last Medieval Queens, which discusses the funeral rites of Anne and other queens, mentions only Crowland's one-sentence account of Anne's funeral.
Caroline Halsted, who wrote a nineteenth-century biography of Richard III, does mention "the tears which [Anne's] husband is allowed to have shed when personally attending her remains to St. Peter's, Westminster." Alas, when one follows the reference Halsted gives, it leads to Richard Baker's Chronicle of the Kings of England (1670). Baker, however, has Richard shedding only "formal tears," not tears of sorrow: "for within few dayes after, whether by poyson, or by what other means, it is not certainly known, she departed this life; and with all solemnity, not without some formal tears of King Richard, was interred in St.* Peter's Church at Westminster."
George Buck, a seventeenth-century apologist for Richard, writes (in the edition of his book prepared by Arthur Kincaid) simply that Richard "was rather taken to be uxorious than otherwise, and at her death expressed it in his heavy mourning, causing very magnificent exequies to be prepared for her, interring her non cum minore honore quam reginat decuit, as the Prior of Croyland testifieth." No funeral tears, no shutting himself up.
Finally, I checked Clements Markham's 1906 Richard III: His Life and Character, and found that Markham does indeed state that Anne "was buried in Westminster Abbey; her sorrowing husband shedding tears over her grave." But Markham cites Buck as his source--and Buck doesn't say that Richard was present at the funeral or was seen to shed tears there. Thinking that the original edition of Buck, the text of which is notoriously corrupt, might bear out Markham's claim, I checked a facsimile of the 1647 edition of Buck cited by Markham and found only this: "he was rather thought uxorious than otherwise; which appeared unfeignedly at her death, in the expression of sorrow and magnificent Exequies for her." Again, this neither places Richard at Anne's funeral nor has him isolating himself; it simply has him grieving and arranging a magnificent funeral.
So, no contemporary source places Richard at Anne's funeral shedding tears of sorrow; this seems to be Halsted's and Markham's embellishment, unsupported by the seventeenth-century sources they cite. And where the story that Richard III shut himself up for three days came from, I still haven't a clue.
Would Richard have even attended Anne's funeral, for that matter? Edward I and Richard II attended their queens' funerals (I haven't figured out whether Edward III was present at Queen Philippa's), but Henry VII, who's known to have grieved after Elizabeth of York died, didn't attend her funeral, and Henry VIII didn't attend Jane Seymour's funeral even after she presented him with his long-sought-after son. It depends, I guess, on whether Richard III followed what seems to have been the earlier custom of presence or the later custom of absence.
The fact that there's no contemporary evidence that Richard attended Anne's funeral, wept for her there, and shut himself up for three days doesn't, of course, mean that he didn't mourn her, though it does leave us only with his word that he did. Nor should it be taken as evidence that he hastened her death, which I've never believed. It does show, however, how unsupported assertions gradually acquire the status of historical fact and, thanks to the Internet, gain vigorous new life.