Tjyylli

In open air, when vapour pressure reaches 1 atm, boiling takes place.

I read that if we add two immiscible liquids together, the total vapour pressure of the 'mixture' is close to $p = p^*_A + p^*_b$, means that the vapour pressure of the 'mixture' is higher than the vapour pressure of both the constituent liquids, subsequently means that the 'mixture' boils at lower temperature than both of the constituent liquids.

If we have many liquids which are not immiscible with each other, and we add them together to form a 'mixture' such that $p = p^*_A + p^*_B + ... > pu{1atm}$ , will boiling results at room temperature?

I suspect we have many liquids which are not miscible with each other, since we classify solvents as organic solvent and inorganic solvent only, but let's say we indeed have that (or we have two immiscible liquids volatile enough to make $p>pu{1atm}$), what will be observed? Boiling at room temperature? In that case, we can use the vapour to do work (e.g. rotate turbine), but when we mix immiscible liquid together there should be no energy transaction right? The particles do not react with each other. If all these are true, we are actually doing work without any input which violates the first law, so there must be something wrong with my reasoning above.

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. 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.

Nothing special would happen, immiscible liquids would just form layers. As for the expression, $$p_T=Sigma p^o_i$$

I suggest you read this answer. Quoting Ivan Neretin:

This is not just some vapor pressure. This is the equilibrium vapor pressure. Thermodynamics is all about equilibrium, you know. And equilibrium, roughly speaking, is what takes place in a closed container after a billion years.

It would take infinite time for that expression to prove itself correct.

In speaking within an average man's lifespan, the liquids would remain in separate layers, with the pressure above liquid surface would remain approximately the pure liquid vapour pressure ($p^o$) of the uppermost layer if left undisturbed (according to your question: no energy transactions).

Don't worry about the first law: it is rarely violated these days.

This "equilibrium" can be attained sooner by agitation (stirring) of the mixture, but mechanical stirring certainly counts as work done.

From a chemguide page:

Obviously if you have two immiscible liquids in a closed flask and keep everything still, the vapour pressure you measure will simply be the vapour pressure of the one which is floating on top. There is no way that the bottom liquid can turn to vapour. The top one is sealing it in.

Agitated mixtures of immiscible liquids will boil at a temperature lower than the boiling point of either of the pure liquids. Their combined vapour pressures are bound to reach the external pressure before the vapour pressure of either of the individual components get there.

In other words, although less evident, even if you do get the mixture to boil and do work, it would be because you had done work on it earlier during agitation.

Is it possible to boil a liquid by just mixing many immiscible liquids together?

No*, boiling is when the vapor pressure of a phase is greater than the ambient temperature. You might create a total pressure of all of the partial pressures in excess of ambient, but it wont cause boiling since the partial pressure of each of the phases is less than ambient, no boiling will occur.

To get boiling by mixing you need two mixable substances that form a positive azeotrope when mixed and are at a sufficiently high temperature that the mixture can boil. Take for example methanol and chloroform system in the phase diagrams below. At $pu{330K}$ neither pure methanol nor pure chloroform boil, but if you mixed $pu{2 mol}$ of chloroform at $pu{330K}$ with $pu{1 mol}$ of methanol at $pu{330K}$ together the phase would boil upon mixing as the vapor pressure of the phase is now above ambient.

Okay I said no, but that technically isn't correct. Grease fires are a (too) common example of mixing immiscible liquids and causing boiling as the oil temperature is well above the boiling temperature of water. This is why you can't throw water on a grease fire. For perspective on thanksgiving of 2016 there were about 1570 fires in the US related to this phenomenon.