LM324 instead of LTC6081 for thermocouple
Maybe, but it will probably not perform as good. As the datasheet of the LT1025 tells us in "Amplifier considerations":


  Thermocouple amplifiers need very low offset voltage and
  drift, and fairly low bias current if an input filter is used.


The LM324 has typically 2mV offset voltage (instead of 70µV of the LTC6081), and a typical temperature drift of 7µV/K (compared to 0.8µV/K maximum on the LTC6081). Also the bias current with 45nA is on the higher side (pA on the LTC6081 and &lt;1nA on others).

But it also depends on your application, will the temperature of the amplifier stay mostly constant? Then temperature drift of the offset voltage isn't that much of a problem.

(Are you using S or R type thermocouples? Then you will be in a world of hurt with the LM324. As pointed out, the schematic assumes a K type thermocouple, thanks @RespawnedFluff)

You can get around the offset if you do some offset correction.

If you can get the LT1025, then maybe some of the recommended amplifiers are available.

Chopper stabilized amplifiers work quite well for thermocouples. Something like the TLC2652A or OPA2735 if TI parts are available.