Is it possible to boil a liquid by just mixing many immiscible liquids together?
Yes they will boil all right. Sure, there might be some kinetic impediment to it if you let the liquids to settle in layers, but if you stir them so as to expose their surfaces, they will boil&dagger;. This is what steam distillation is all about.

As for the first law, it will hold just fine. You burn your firewood, you get the heat, but it is not for free: the firewood is gone. Same thing here. Your liquids are gone. What you have now is a vapor mixture. Sure, you may cool it down, and it will separate back into the liquids, which you may then heat up and repeat... Congratulations, you've just invented the steam engine. Pity it's been done before by one James Watt. Also, you don't really need two liquids for it.

So it goes.



(&dagger;) There seem to be some ambiguity as to what exactly "boiling" entails.


Will vapor bubbles form in the whole bulk of both liquids? No.
Will both liquids eventually evaporate completely, if left in a cylinder under a piston and maintained at these pressure and temperature until they reach equilibrium? Yes.


For a single pure liquid these two questions are equivalent, hence the confusion.