Can I drive an LS on the stock tune after adding headers and a full exhaust?

Usually yes, to drive it — but you're leaving power and drivability on the table.

• The general rule: LSX Magazine's rule is that any change that significantly alters airflow or fuel flow (injectors, cam, throttle body) needs a custom calibration, because the stock fuel and timing data no longer match the engine.
• Bolt-ons specifically: Summit's LS upgrade guides treat intake, exhaust and similar bolt-ons as garage jobs that run, but note the tune won't be optimized. Their advice is to follow them with a chassis-dyno session. On a stock LS6, Summit puts basic bolt-ons plus tuning at about 340 → 400 whp.
• If you remove emissions parts (cats, rear O2 sensors, EGR, air pump), the engine can run without them, but the computer may need reprogramming so it doesn't set trouble codes for the missing equipment.

A sensible plan: install the parts, drive it gently, and book a tune. Any check-engine light or odd fuel trims are a reason to tune sooner.

Want to do it yourself? See who tunes it, and how to learn.