The significant question is whether such a statement as "no copyright infringement intended" will be viewed by the courts as evidence to suggest innocent infringement or rather as evidence to support willful infringement. This consideration of willful versus innocent is relevant when a copyright holder seeks statutory damages.
17 USC §504(c)(2) says (emphasis mine):
In a case where the copyright owner sustains the burden of proving, and the court finds, that infringement was committed willfully, the court in its discretion may increase the award of statutory damages to a sum of not more than $150,000. In a case where the infringer sustains the burden of proving, and the court finds, that such infringer was not aware and had no reason to believe that his or her acts constituted an infringement of copyright, the court in its discretion may reduce the award of statutory damages to a sum of not less than $200.
Branch v. Ogilvy & Mather, Inc. says of willfulness (emphasis mine):
Thus, in order to prove willfulness, a plaintiff must show that the defendant knew or should have known that its conduct constituted copyright infringement. Similarly, in order to establish innocent intent, the defendant must prove that it did not know and should not have known that its conduct constituted infringement. One defense to an allegation of willfulness is the defendant's reasonable and good faith belief that its actions do not constitute copyright infringement.
The central question, then, seems to be whether the notice "no copyright infringement intended" aids or harms an argument that the infringer "did not know or should not have known" that they were infringing copyright, or else that they had a "reasonable and good faith belief" of non-infringement.
I am not aware of any case law that concerns such a notice, but my layman intuition suggests to me that
such a notice lends evidential weight that the defendant did indeed know the work in question was under copyright;
since the defendant was therefore aware the work was under copyright, it is unlikely that they could successfully claim a reasonable belief that publicly distributing the entirety of (or a substantial portion of) the work without permission was not infringement.
Ultimately, I don't think such a notice does anything to aid the defendant's argument that they "should not have known" their use was infringing. Nor do I think it furthers an argument of "reasonable and good faith belief" by the defendant: the reasonableness of such a belief will rest significantly on the facts of the infringement, not only on how loudly they shouted, "I swear this isn't infringement!" Conversely, such a notice may further the idea that the defendant did know that copyright applied to the work, strengthening the plaintiff's argument that the defendant should have known the use was infringing.