Is it permissible to place a noble Quranic verse, such as: "My Lord, increase me in knowledge," in the preface of a university doctoral dissertation, knowing that it might be borrowed by non-Muslim students?

There is no objection to placing the aforementioned verse at the beginning of a research paper or thesis, as long as it is in a respectable place and your intention is as you indicated. Your choice of this verse as an introduction to your thesis is very appropriate, for Islam encourages knowledge and reading, and highlights the people of knowledge and their virtue. This verse contains evidence of the virtue of knowledge and the honor of scholars. Non-Muslims becoming aware of it can invite them to learn about the teachings of Islam.