Is power balance a relevant factor in the consumption of a house?
Is this correct? 


Yes, if you overload one phase, then that phase would get hotter, which would increase its resistance, which would decrease the amount of power you get per Ah, which would make a kW more expensive for you.

That being said: won't happen, because copper is an excellent conductor, and it's unlikely you can increase the temperature of a phase sufficiently to make that effect measurable, unless you really try extremely hard, by putting all large consumers on the same phase.

It's more likely that the simple formula \$P_\text{cable loss}=U\cdot I = (I\cdot R_\text{cable})\cdot I = I^2 R_\text{cable}\$ means that it's cleverer to have three small currents on three different phases than one large current on a single phase – safely assuming \$R\$ is constant. I don't know why their advertising didn't go with that:

\begin{align}
P_\text{all current on single phase} &amp;=\\ 1\cdot I_\text{total}^2 R_\text{cable}
&amp;&lt;\frac13 \cdot I_\text{total}^2 R_\text{cable}\\
&amp;= 3\cdot \left(\frac{I_\text{total}}{3}\right)^2 R_\text{cable}\\
&amp;=P_\text{current distributed equally over 3 phases}\\
\end{align}

So, by distributing your loads over three phases evenly, you can save two thirds of the losses in your cable. 

Is that relevant to your bill:


  Is balancing the power load relevant for a domestic electrical bill?


No. If you're an industrial-sized customer, you'll be required to do something like that, but for you, the difference in bill will be 0 to negligible.

Your installation will not be absurdly unbalanced. If it is, doesn't take a consultant to figure out.


  I've heard this might become more relevant in very hot places where cables get hot naturally, is this true?


Um, no, due to the effect being so negligible.

There's a globally uneasy correlation between heat and quality of infrastructure, and that will be way more important here: if your power lines are underdimensioned, then this will be much more of a problem. Not that it's likely to be more of a problem than the power outages in a grid that's under perpetual use larger than it's been designed for.


  There is a company advertising a power balancing service. 


Ha! We can save you a bit of money on power balancing services, then! Don't connect all the heavy users of power (heating, water warming, cooking, larger pumps, dryers, large fridges) on the same phase, but distribute them across different phases. 

Your electrician typically likely did that "by ear" when he wired up the fuses in your house: first room, first phase; second room, second phase; third room, third phase; fourth room, first phase… and that statistically solves the issue quite well. Many heavy duty devices (ovens, water heaters) are themselves designed to be 3-phase devices, meaning there's nothing to optimize there.

Boom! put a superfluous service provider out of business.