Three-dimensional (3D) bioprinting is increasingly used to construct patient-relevant tumor models, but its value for precision oncology depends on how functional biomaterials connect tissue reconstruction with therapeutic exposure and response measurement. This review presents a function-oriented framework for advanced bioprinting biomaterials in personalized cancer therapy and drug screening. We define functional biomaterials as printable or print-compatible systems that actively regulate cellular behavior, extracellular-matrix cues, transport, release, microenvironmental adaptation, or response reporting, while distinguishing them from patient-derived biological inputs and enabling technologies. We examine how bioink composition, crosslinking, mechanics, degradation, and spatial patterning influence tumor heterogeneity, stromal interactions, vascular, immune, and hypoxic niches, and therapeutic transport. Particular attention is given to localized drug delivery, nanocomposite and stimuli-responsive systems, sequential exposure, and quantitative evaluation of drug distribution and response. Two representative platforms illustrate how material design can couple microenvironment reconstruction with chemoradiotherapy, chemotherapy, and chimeric antigen receptor-T cell response testing. We also assess patient-derived tumor models for personalized drug testing, including model establishment, heterogeneity preservation, assay reproducibility, and the relationship between ex vivo response and clinical decision-making. Bioink-selection principles and study-level reporting benchmarks are summarized without pooling non-equivalent metrics. Finally, we discuss sample attrition, turnaround time, cost, manufacturing variability, quality control, interlaboratory reproducibility, clinical validation, and regulatory requirements. Clinically useful platforms will be those in which each functional component addresses a defined biological or therapeutic question and generates reproducible, interpretable information, rather than those with the greatest material or architectural complexity.