Does any scholarship require the benefiting student to make open access any research paper they wrote during the scholarship period?
The Bill &amp; Melinda Gates Foundation funds scholarships and has an open access policy:


  [...] We have adopted an Open Access policy that enables the unrestricted access and reuse of all peer-reviewed published research funded, in whole or in part, by the foundation, including any underlying data sets. [...]


The Max Planck Society in Germany funds scholarships and has an open access policy. 

These are the two examples I remembered directly.