Many philosophers have argued that you should only assert what you know to be the case (e.g. Williamson 1996). If you don't know that P is true, you shouldn't go around saying that P is true. Furthermore, to assert what you don't know isn't just bad manners; it violates a constitutive norm, fundamental to what assertion is. To accept this view is to accept what's sometimes called the Knowledge Norm of Assertion.
Most philosophers also accept the view, standard in epistemology, that you cannot know something that you don't believe. Knowing that P implies believing that P. This is sometimes called the Entailment Thesis. From the Knowledge Norm of Assertion and the Entailment Thesis, the Belief Norm of Assertion follows: You shouldn't go around asserting what you don't believe. Asserting what you don't believe violates one of the fundamental rules of the practice of assertion.
However, I reject the Entailment Thesis. This leaves me room to accept the Knowledge Norm of Assertion while rejecting the Belief Norm of Assertion.
Here's a plausible case, I think.
Juliet the implicit racist. Many White people in academia profess that all races are of equal intelligence. Juliet is one such person, a White philosophy professor. She has studied the matter more than most: She has critically examined the literature on racial differences in intelligence, and she ﬁnds the case for racial equality compelling. She is prepared to argue coherently, sincerely, and vehemently for equality of intelligence and has argued the point repeatedly in the past. When she considers the matter she feels entirely unambivalent. And yet Juliet is systematically racist in most of her spontaneous reactions, her unguarded behavior, and her judgments about particular cases. When she gazes out on class the ﬁrst day of each term, she can’t help but think that some students look brighter than others – and to her, the Black students never look bright. When a Black student makes an insightful comment or submits an excellent essay, she feels more surprise than she would were a White or Asian student to do so, even though her Black students make insightful comments and submit excellent essays at the same rate as the others. This bias affects her grading and the way she guides class discussion. She is similarly biased against Black non-students. When Juliet is on the hiring committee for a new ofﬁce manager, it won’t seem to her that the Black applicants are the most intellectually capable, even if they are; or if she does become convinced of the intelligence of a Black applicant, it will have taken more evidence than if the applicant had been White (adapted from Schwitzgebel 2010, p. 532).
Does Juliet believe that all the races are equally intelligent? On my walk-the-walk view of belief, Juliet is at best an in-between case -- not quite accurately describable as believing it, not quite accurately describable as failing to believe it. (Compare: someone who is extraverted in most ways but introverted in a few ways might be not quite accurately describable as an extravert nor quite accurately describable as failing to be an extravert.) Juliet judges the races to be equally intelligent, but that type of intellectual assent or affirmation is only one piece of what it is believe, and not the most important piece. More important is how you actually live your life, what you spontaneously assume, how you think and reason on the whole, including in your less reflective, unguarded moments. Imagine two Black students talking about Juliet behind her back: "For all her fine talk, she doesn't really believe that Black people are just as intelligent."
But I do think that Juliet can and should assert that all the races are intellectually equal. She has ample justification for believing it, and indeed I'd say she knows it to be the case. If Timothy utters some racist nonsense, Juliet violates no important norm of assertion if she corrects Timothy by saying, "No, the races are intellectually equal. Here's the evidence...."
Suppose Tim responds by saying something like, "Hey, I know you don't really or fully believe that. I've seen how to react to your Black students and others." Juliet can rightly answer: "Those details of my particular psychology are irrelevant to the question. It is still the case that all the races are intellectually equal." Juliet has failed to shape herself into someone who generally lives and thinks and reasons, on the whole, as someone who believes it, but this shouldn't compel her to silence or compel her to always add a self-undermining confessional qualification to such statements ("P, but admittedly I don't live that way myself"). If she wants, she can just baldly assert it without violating any norm constitutive of good assertion practice. Her assertion has not gone wrong in a way that an assertion goes wrong if it is false or unjustified or intentionally misleading.
Jennifer Lackey (2007) presents some related cases. One is her well-known creationist teacher case: a fourth-grade teacher who knows the good scientific evidence for human evolution and teaches it to her students, despite accepting the truth of creationism personally as a matter of religious faith. Lackey uses this case to argue against the Knowledge Norm of Assertion, as well as (in passing) against a Belief Norm of Assertion, in favor of a Reasonable-To-Believe Norm of Assertion.
I like the creationist teacher case, but it's importantly different from the case of Juliet. Juliet feels unambivalently committed to the truth of what she asserts; she feels no doubt; she confidently judges it to be so. Lackey's creationist teacher is not naturally read as unambivalently committed to the evolutionary theory she asserts. (Similarly for Lackey's other related examples.)
Also, in presenting the case, Lackey appears to commit to the Entailment Thesis (p. 598: "he does not believe, and hence does not know"). Although it is minority opinion in the field, I think it's not outrageous to suggest that both Juliet and the creationist teacher do know the truth of what they assert (cf. the geocentrist in Murray, Sytsma & Livengood 2013). If the creationist teacher knows but does not believe, then her case is not a counterexample to the Knowledge Norm of Assertion.
A related set of cases -- not quite the same, I think, and introducing further complications -- are ethicists who espouse ethical views without being much motivated to try to govern their own behavior accordingly.