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The essence of precision oncology lies in its commitment to tailor targeted treatments and care measures to each patient based on the individual characteristics of the tumor. The inherent heterogeneity of tumors necessitates gathering information from diverse data sources to provide valuable insights from various perspectives, fostering a holistic comprehension of the tumor. Over the past decade, multimodal data integration technology for precision oncology has made significant strides, showcasing remarkable progress in understanding the intricate details within heterogeneous data modalities. These strides have exhibited tremendous potential for improving clinical decision-making and model interpretation, contributing to the advancement of cancer care and treatment. Given the rapid progress that has been achieved, we provide a comprehensive overview of about 300 papers detailing cutting-edge multimodal data integration techniques in precision oncology. In addition, we conclude the primary clinical applications that have reaped significant benefits, including early assessment, diagnosis, prognosis, and biomarker discovery. Finally, derived from the findings of this survey, we present an in-depth analysis that explores the pivotal challenges and reveals essential pathways for future research in the field of multimodal data integration for precision oncology.