Can you claim ink for a printer as a school expense?

Probably not. Education expenses need to be required by the institution.

These expenses qualify, as long as they are required to enroll in or attend a qualified educational institution:

• Tuition, if paid to the institution
• Student-activity fees, if paid to the institution
• Books, supplies, and equipment, if required to be purchased from the institution as a condition of enrollment or attendance (Lifetime Learning Credit)
• Books, supplies, and equipment, regardless of where purchased (American Opportunity Credit)

So, I can claim my required books that I purchased from outside of the school? What about lab notebooks that were required but not sold by the school?

Yes, if you are claiming the American Opportunity Credit then you can claim the required supplies, like books and notebook, purchased elsewhere. As ColeenD3 stated "Books, supplies, and equipment, regardless of where purchased (American Opportunity Credit)" qualify.