For the most part, under US law, whether a use is infringement or not depends on the facts of the case, and not on the intent of the alleged infringer. Intent matters when a use is found to be infringing, and the copyright owner elects to receive statutory damages rather than actual damages. The court has a wide range of discretion under 17 USC 504 (c). The base range (see 17 USC 504(c)(1)) is anywhere from $750 to $30,000 per work infringed (not per infringing copy made) "as the court considers just". If the plaintiff alleges and proves that the infringement was "willful" then the upper limit on statutory damages is $150,000, but the court still has authority to fix the damages anywhere in the range. If the defendant claims, and proves, "that such infringer was not aware and had no reason to believe that his or her acts constituted an infringement of copyright" the lower limit is reduced to $200, but the court sill has discretion to fix the amount of damages at any sum up to $30,000 (this is "innocent infringement"). A reasonable belief that the use was a fair use comes into play only when the infringer is "an employee or agent of a nonprofit educational institution, library, or archives acting within the scope of his or her employment" or is "a public broadcasting entity" or someone working with or for such an entity. In those limited cases statutory damages are not imposed. (These adjustments are spelled out in 17 USC 504(c)(2).)
Note that the mere presence of a copyright notice is generally enough to defeat a claim of innocent infringement under 17 USC 401(d). Thus even here "intent" is of relatively little import. What matters is what an infringer knew or reasonably should have known, or reasonably believed. There is a gap between "innocent" and "willful" infringement. Many careless infringements will be neither. But when there is a notice saying "no copyright infringement intended" that tends to suggest that the infringer knew what copyright infringement is, and knew that the content was or might be protected by copyright, which hurts any claim for innocent infringement. The only plausible positive effect of such a notice that I can see is if it induces a copyright owner not to sue in the first place, as it might. Such a notice might possibly support an argument that the infringement, while not "innocent" was not "willful". But otherwise the intent, however honest, of the infringer to rely on fair use defense does not legally change the fact that the infringer knew the content was protected by copyright, and thus the infringement was not "innocent". Th facts supporting such a claim of "fair use intent" might influence the court in deciding whether the infringement was "willful" and might further influence the court in determining what amount of statutory damages is "just". But the law does not specify that an intent to rely on fair use be even taken into account in the decision on the amount of statutory damages.