Is the following a form of dishonesty on the part of the school? This distance-learning school claims to be a member of a certain international academic organization . But after checking with the said organization, I found out that the school's membership have expired years ago. Just last week, the organization informed me that they have requested the school to remove the questioned information from its website. To date, however, the school has not yet acted on this request. 

Can the school's action or inaction qualify as a form of dishonesty or should we give the school the benefit of the doubt? 

(By the way, the school is government-recognized/accredited.)
Does no really matter if the school is government recognized or not. Two things here. 

Accreditation / Membership -  usually costs a lot of money each time you need to "refresh" it, so the school may have chosen not to do so. That does not necessarily mean, the school is not up to the standard. It may even be that school has decided not to participate as a member to this organization because of reasons unknown to you. 

Next to that, these companies makes its name by doing proper evaluations. Those are always based on certain time period, and the conditions in the school at that time. Within 4 year period, a school can change half of its staff, have a new director, and a completely different set of standards, guidelines, and internal quality assessments. This is why schools must comply with the accreditation or membership standards continuously in order to keep their accreditation. And that is why these companies have a right to ask the school to remove it from their website / media. 

It is not possible to determine from your writing, if we can attribute the school with dishonesty (requires knowledge of wrongdoing) or if this is part of the school policies working too slow, but in any case, if the company has contacted them, and instructed them to remove the false media materials, and they do not do so in a reasonable time, I would consider it at least a breach of trust, and most likely also contract. 

Please note, that for a school that most likely advertises every start of the academic year, it will most likely make financial and moral sense to change these materials shortly before that start of the new academic year, which is a long time. If this bothers you, always feel free to write a letter to your school explaining your viewpoint. You can also write anonymously.