Both proofs in the question are of the following type. Suppose you want to prove a certain identity among numbers $X$, say $f(X)=0$. If you can find another function $g(X,Y)$ such that $f(X)=g(X,Y)$ for some $Y$. Then, proving that $g(X,Y)=0$ for this choice of $Y$, independent of $X$, automatically proves $f(X)=0$.
In the case of the Pythagorean theorem, $X=(a,b,c)$ are the dimensions of the triangular base of the water tank, $Y$ are all the other details of the physical setup and $g(X,Y)$ is the total torque about $BB'$ exerted on the tank. As long as $Y$ contains the fact that there are no torques acting on the tank other than the different parts of the tank acting on each other, physical law dictates that the total torque must be zero, hence $g(X,Y)=0$. The Kut-the-knot page linked from one of jc's comments shows why $g(X,Y)$ is proportional to $c^2-a^2-b^2$.
The case of the tetrahedron is similar, $X$ are the dimensions of the tetrahedron and $g(X,Y)$ is the total force acting on the gas container. The physical situation $Y$ is set up such that no forces act on the container external to the container itself, hence physical law dictates that the total force must be zero.
Now, what is it that allows us to conclude something about the value of $g(X,Y)$ in each case, given only $Y$ and independent of $X$? One way to think about it is to consider the physical world as a giant computer or oracle, $W$. If you set up a physical situation $Z$ and measure some observable quantity $G$, the world will give you a numerical answer $W(G,Z)$. The goal of physics, of course, is to build mathematical models of $W$, so that we don't have to consult the oracle every time we need an to know the value of $W(G,Z)$. But a mathematical model is not necessary if we are happy with just consulting the oracle when needed. Still, observation of the answers that we get from $W$ allows us to identify certain regularities in its output: these are the laws of physics.
Thus, supposing that we can find $G$ and $Z=(X,Y)$ such that $g(X,Y)=W(G,Z)$, the kind of of proof I outlined in the first paragraph can be carried out, appealing to the laws of physics as properties of the oracle $W$. To convert this kind of proof into a sequence of logical deductions, as a usual proof should be, requires a mathematical model for $W$, at least a restricted one that suffices for the purposes of making deductions about $g(X,Y)$. In both cases the model that does the job is plain old Newtonian mechanics of particles, solid bodies and fluids (if desired, solid bodies and fluids may be seen as a limiting cases of large assemblies of particles). The equality $g(X,Y)=f(X)$ follows from direct application of Newton's second law (an axiom of this mathematical model) and the identity $g(X,Y)=0$, independent of $X$, follows from Newton's third law (another axiom).
Personally, I think the appeal of such "physical" proofs is the fact that they rely only on some properties of $W$, and not on the details of a mathematical model for it. However, the validity of the proof only follows if one can show that a detailed mathematical model exists.
Hope this helps.
A weak form of your BVP is $a(q,v)=\ell(v)$ where $a(q,v)=\int_{\Omega}\nabla q\cdot\nabla v\,dx+\int_{\Omega}\beta qv\,dx$ and $\ell(v)=\int_{\Omega}fv\,dx+\int_{\partial\Omega}gv\,ds$ with $q,v\in H^1(\Omega)$.
If $\beta>0$, the bilinear form is coercive and continuous in $H^1(\Omega)$. Thus, apply Lax-Milgram and you get existence, uniqueness and stability in $H^1(\Omega)$. Stability now depends on the value of $\beta$.
Best Answer
Well, I don't understand the electrostatics, but here is another physical heuristic:
Impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior. This gives a harmonic function extending the exterior temperature. [What's the electrostatic analogue? Formerly I had written "charge density", but now I am not sure if that's right.]
I think this strongly suggests a mathematically rigorous argument: We are naturally led to model the time-dependence of temperature in the interior. This satisfies a diffusion (or heat) equation, but in words:
"After a time \delta, the new temperature is obtained by averaging the old temperature along a circle of radius \sqrt{\delta}."
This process converges under reasonable conditions, as time goes to infinity, to the solution of the Dirichlet problem. Anyway, we are led to the Brownian-motion proof of the existence, which I personally find rather satisfying. Another personal comment: I think one should always take "physical heuristics" rather seriously.
[In response to Q.Y.'s comments below, which were responses to previous confused remarks that I made: neither the electric field nor the Columb potential is a multiple of the charge density on the boundary: the former is a vector, and in either case imagine the charge on the boundary to be concentrated in a sub-region; neither the electric field nor the potential will be constant outside that sub-region.]