How to Amplify Thermocouples (and DC sensors between 1mV and 1µV DC)?
No idea, maybe you applied reverse voltage to the supply terminals. 
You can use 0V if you give the op-amp +/- supplies. The one you are using requires at least -4V on the negative supply to work near 0V on the inputs. See the data sheet where it talks about "common mode voltage range". The circuit you show has no negative supply so you are not going to get useful results. It still should not damage the op-amp. 
The op-amp you are using may be useful for some things, but it's about useless as a thermocouple amplifier. Note the large offset voltage (+/-5mV typical,  up to +/-13mV). That could be trimmed out, but the offset voltage change  with temperature is a typical 18uV/K (no maximum is given), which means you will have big compensation errors. Chopper (zero drift) op-amps are much better, you can get relatively inexpensive op-amps with drifts of 50nV/K which is several hundred times better. The offset voltage would also be a fraction of a °C equivalent. 
Thermocouples work well when the junction is grounded, but it's not necessary, if you isolate (galvanically) the junction you can ground the negative terminal of the thermocouple. Of course you don't want to thermally isolate the junction or it won't follow whatever it is you are measuring. Note that heat travels through the wires so there is work being done to maintain a temperature differential- this is the thermodynamic basis of the thermocouple- you are not getting voltage (and potentially current) from nothing. 


With a better op-amp and dual supplies you can get useful results, though you'll have to supply cold junction compensation in some manner (for example by measuring the temperature where the thermocouple metals are attached to copper) and compensating digitally, if you want decent accuracy.