Deploying large language models (LLMs) as cloud services raises privacy concerns as inference may leak sensitive data. Fully Homomorphic Encryption (FHE) allows computation on encrypted data, but current FHE methods struggle with efficient and precise nonlinear function evaluation. Specifically, CKKS-based approaches require high-degree polynomial approximations, which are costly when target precision increases. Alternatively, TFHE's Programmable Bootstrapping (PBS) outperforms CKKS by offering exact lookup-table evaluation. But it lacks high-precision implementations of LLM nonlinear layers and underutilizes GPU resources. We propose \emph{TIGER}, the first GPU-accelerated framework for high-precision TFHE-based nonlinear LLM layer evaluation. TIGER offers: (1) GPU-optimized WoP-PBS method combined with numerical algorithms to surpass native lookup-table precision limits on nonlinear functions; (2) high-precision and efficient implementations of key nonlinear layers, enabling practical encrypted inference; (3) batch-driven design exploiting inter-input parallelism to boost GPU efficiency. TIGER achieves 7.17$\times$, 16.68$\times$, and 17.05$\times$ speedups over a CPU baseline for GELU, Softmax, and LayerNorm, respectively.

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