Is my LM35 not working?
The data sheet sample circuit shows either an open output (presumably into a high impedance input such as the '328's A/D), or connected to -Vs via a resistor chosen to draw 50 mA. Access to -Vs is not typical of our applications but they do show a current-limiting resistor to ground in that case. So I'd expect that grounding the output caused the device to draw more current than it could dissipate without overheating.