Can we think of the EM tensor as an infinitesimal generator of Lorentz transformations?
Physically, the only thing that the electromagnetic field tensor and a Lorentz transformation generator have in common is that they both happen to be antisymmetric rank 2 tensors. The link doesn't go any farther than that.
However, this coincidence does lead to a few analogies. For example, if you know about Lorentz transformations, then you know that an antisymmetric rank 2 tensor contains two three-vectors inside it, namely $\boldsymbol{\zeta}$ and $\mathbf{K}$. Then if somebody tells you the electromagnetic field is the same kind of tensor, you'll automatically know that it can be broken down into two three-vectors, namely the electric and magnetic fields. But this is a purely mathematical analogy.
A more physical result comes from the equation of motion $$\frac{d u_\mu}{d\tau} = (q/m) F_{\mu\nu} u^\nu.$$ where $u^\mu$ is the four-velocity; you can expand this in components to verify it's just the Lorentz force law. Now, comparing this with an infinitesimal (active) Lorentz transformation $$\Delta u_\mu = \Lambda_{\mu\nu} u^\nu$$ we see that the Lorentz force is equivalent to an active Lorentz transformation acting on the four-velocity, with generator $(q/m) F_{\mu\nu}$.
We can do some quick sanity checks:
- Magnetic fields cause rotations. If we start with a nonzero three-velocity and apply a magnetic field, the velocity spins around.
- Electric fields cause boosts. If we apply an electric field, the three-velocity grows in the direction of the field, just like it does in the direction of a boost.
Two caveats to this result:
- As stated in the link you gave, this result doesn't allow us to think of electromagnetism as a geometric phenomenon, because different particles have different values of $q/m$ and hence are acted on by different Lorentz transformations. It's just a nice heuristic.
- Be careful to distinguish between active and passive Lorentz transformations. Most of the ones you'll run into are passive (i.e. used to switch between coordinate systems), but as ACM points out, such transformations are described by matrices, not tensors. Above, I'm considering active rotations and boosts, and everything is taking place in a single coordinate system.
The electromagnetic field strength tensor is not a Lorentz generator.
First, even when written in matrix form, the signs are wrong. The boost generators are of the form $$ \begin{pmatrix} 0 & v_x & v_y & v_z \\ v_x & 0 & 0 & 0 \\ v_y & 0 & 0 & 0 \\ v_z & 0 & 0 & 0 \end{pmatrix},$$ which are not antisymmetric.
Second, the EM tensor is not a matrix, it's a 2-form $F = F_{\mu\nu}\mathrm{d}x^\mu \wedge\mathrm{d}x^\mu$, while the Lorentz generators are actual matrices, not coefficients of a form. Writing $F_{\mu\nu}$ as a matrix does not reflect its geometric nature. It does not generate Lorentz transformations, it itself transforms under them as an ordinary 2-tensor.