To protect his sweetheart, Becky, from punishment in school, what does Tom lie about in Adventures of Tom Sawyer?

Tom lies about the teacher's book.  Becky had accidentally torn it, but Tom takes the blame.

Mr. Dobbins, the "master" at school, has a "mysterious book" which he keeps in his desk and takes out and reads at times during the day "when no classes (are) reciting.  Mr. Dobbins had aspired to be a doctor, but, having had no money for medical school, had ended up being "nothing higher than a village schoolmaster" instead.  The book in Mr. Dobbins' desk addresses the subject of anatomy, a subject in which the teacher would have been well-versed had he been able to pursue his true ambition.  When he has free time during the school day, he takes the opportunity to absorb himself in the study of an area that he loves.

The students in Mr. Dobbins' class have all caught glimpses of the book, but no one knows what kind of book it is.  On the day in question, Becky is alone by the teacher's desk, and notices that the key is in the lock.  Succumbing to curiosity, she opens the desk and takes the mysterious book out for examination.  As she is looking at it, Tom comes by; Becky closes the book and puts it away, tearing one of the pages.

When Mr. Dobbins discovers that his book has been damaged, he immediately confronts the class, asking the students individually if he or she is the culprit.  One by one the students deny having torn the book, but when Mr. Dobbins begins to question Becky, Tom knows that the teacher will discern her guilt by the look on her face.  Precipitously, Tom leaps to his feet and shouts, "I done it!"

Tom takes Becky's punishment, "the most merciless flaying that even Mr. Dobbins had ever administered", but to him, "the surprise, the gratitude, the adoration that (shines) upon him out of poor Becky's eyes (seem) pay enough for a hundred floggings".  In addition to being beaten, Tom is forced to "remain two hours after school should be dismissed", but he doesn't mind because he knows when "his captivity (is) done", Becky will be waiting for him outside to thank him (Chapter 20).