Why would Mr. Frank ask for some of Anne's journal entries not to be published?

THere would be too much personal and, what he felt, embracing information.

For more information, please go to the Anne Frank website; I've provided the link below; also, there's a fantastic Masterpiece Classic version of the novel that was made wuite recently. If you have the time to rent and watch, I'd definitely take a look!

Very few passages

Otto Frank was well-read, but he lacked experience in the mores of publishing. In the original printed version of ‘The Diary of a Young Girl’ there is no acknowledgment that he had edited his daughter's writings nor that, in preparing the book, he had time and again made choices between two very different versions of Anne's diary entries. The very brief, unsigned epilogue simply reads: ‘Apart from very few passages, which are of little interest to the reader, the original text has been printed.’ Otto Frank stood by this fiction until the end of his life in 1980. When asked what sections he had left out, he would answer steadfastly that he had eliminated ‘nothing essential, just passages about Anne's physical development and nasty remarks about her mother.’ With this standard reply Otto Frank created a false perception that has survived him to this day. Mirjam Pressler banked on it for her new edition of ‘The Diary of a Young Girl,’ despite the fact that a careful reading of the Critical Edition refutes Otto Frank's assertion.

More self-critical and introspective

It's not easy to second-guess why Otto Frank never let on to the existence of two primary--albeit fragmentary--versions of Anne's diary. Did he think that a quasi-unedited, spontaneous diary of a young girl would have a stronger appeal to prospective readers? What were his guiding principles when, preparing the first edition, he made his choices between a- and b-texts?

For the year 1943, he only had Anne's revised manuscript, the b-version, at his disposal, since the a-version was lost. For that period of time, the reader gets the best insight into the kind of texts Anne had prepared for publication: vignettes about the daily routine in the "back-quarters" and witty descriptions of keenly observed special episodes that interrupt that routine. Anne has kind and appreciative words for the helpers, especially for Miep Gies, but she also reports about the food shortage, about the many unpleasant quarrels in the tense atmosphere of the cramped quarters, and about the news from the outer world, particularly concerning the rounding up of more and more Jews and the progress of the war. Towards the end of 1943 her entries become more self-critical and introspective.

Eliminated text

From this period date the three most striking pages of the five loose sheets that emerged in the nineteen nineties. During the last year of his life, Otto Frank had entrusted these five pages to Cor Suijk, then a staff member of the Anne Frank Foundation. In the meanwhile, a new Critical Edition of ‘The Diary of Anne Frank’ appeared in 2003 (based on the new Dutch edition of 2001), in which these three pages, the b-version of the diary entry of February 8, 1944 are included, together with 47 lines of the original a-version under that same date. The latter had been eliminated in the Critical Edition of 1989 at the request of the Frank family. The passage contains Anne’s reflections about her parents’ marriage.