Can chemo damage a person's heart so much that they are ineligible for it when their leukemia relapses 10 months later?

Unlikely: It's theoretically possible (like just about everything else in the world) but very rare and very unlikely.

Possible: Certain medications used for chemotherapy have significant adverse effects including cardiac toxicity, so what you describe is possible. Usually, if the leukemia comes back, the offending drug is replaced if there are reasonable alternatives.