As Large Language Models (LLMs) become increasingly prevalent in tasks related to cultural heritage, such as generating descriptions of historical monuments, translating ancient texts, preserving oral traditions, and creating educational content, their ability to produce accurate and culturally aligned texts is being increasingly relied upon by users and researchers. However, cultural value misalignments may exist in generated texts, such as the misrepresentation of historical facts, the erosion of cultural identity, and the oversimplification of complex cultural narratives, which may lead to severe consequences. Therefore, investigating value misalignment in the context of LLM for cultural heritage is crucial for mitigating these risks, yet there has been a significant lack of systematic and comprehensive study and investigation in this area. To fill this gap, we systematically assess the reliability of LLMs in generating culturally aligned texts for cultural heritage-related tasks. We conduct a comprehensive evaluation by compiling an extensive set of 1066 query tasks covering 5 widely recognized categories with 17 aspects within the knowledge framework of cultural heritage across 5 open-source LLMs, and examine both the type and rate of cultural value misalignments in the generated texts. Using both automated and manual approaches, we effectively detect and analyze the cultural value misalignments in LLM-generated texts. Our findings are concerning: over 65% of the generated texts exhibit notable cultural misalignments, with certain tasks demonstrating almost complete misalignment with key cultural values. Beyond these findings, this paper introduces a benchmark dataset and a comprehensive evaluation workflow that can serve as a valuable resource for future research aimed at enhancing the cultural sensitivity and reliability of LLMs.