Education technologies (edtech) are increasingly incorporating new features built on large language models (LLMs), with the goals of enriching the processes of teaching and learning and ultimately improving learning outcomes. However, the potential downstream impacts of LLM-based edtech remain understudied. Prior attempts to map the risks of LLMs have not been tailored to education specifically, even though it is a unique domain in many respects: from its population (students are often children, who can be especially impacted by technology) to its goals (providing the correct answer may be less important for learners than understanding how to arrive at an answer) to its implications for higher-order skills that generalize across contexts (e.g., critical thinking and collaboration). We conducted semi-structured interviews with six edtech providers representing leaders in the K-12 space, as well as a diverse group of 23 educators with varying levels of experience with LLM-based edtech. Through a thematic analysis, we explored how each group is anticipating, observing, and accounting for potential harms from LLMs in education. We find that, while edtech providers focus primarily on mitigating technical harms, i.e., those that can be measured based solely on LLM outputs themselves, educators are more concerned about harms that result from the broader impacts of LLMs, i.e., those that require observation of interactions between students, educators, school systems, and edtech to measure. Overall, we (1) develop an education-specific overview of potential harms from LLMs, (2) highlight gaps between conceptions of harm by edtech providers and those by educators, and (3) make recommendations to facilitate the centering of educators in the design and development of edtech tools.