Perfect Rhymes:
* burn my
* turn my
* yearn my (less common, but still works)
Near Rhymes:
* learn why
* learn by
* learn nigh (archaic)
The best rhyme depends on the context of your sentence.
For example, if you're talking about someone's desire to learn, "burn my" might be too intense, while "turn my" could be a good fit:
* "I like to learn my lessons, but sometimes they burn my brain."
* "I like to learn my lessons, and I turn my mind to them every day."
Let me know if you have a specific sentence in mind, and I can help you find the perfect rhyme!