Automated respiratory audio analysis promises scalable, non-invasive disease screening, yet progress is limited by scarce labeled data and costly expert annotation. Zero-shot inference eliminates task-specific supervision, but existing methods apply uniform computation to every input regardless of difficulty. We introduce TRIAGE, a tiered zero-shot framework that adaptively scales test-time compute by routing each audio sample through progressively richer reasoning stages: fast label-cosine scoring in a joint audio-text embedding space (Tier-L), structured matching with clinician-style descriptors (Tier-M), and retrieval-augmented large language model reasoning (Tier-H). A confidence-based router finalizes easy predictions early while allocating additional computation to ambiguous inputs, enabling nearly half of all samples to exit at the cheapest tier. Across nine respiratory classification tasks without task-specific training, TRIAGE achieves a mean AUROC of 0.744, outperforming prior zero-shot methods and matching or exceeding supervised baselines on multiple tasks. Our analysis show that test-time scaling concentrates gains where they matter: uncertain cases see up to 19% relative improvement while confident predictions remain unchanged at minimal cost.