If $F\to E\to B$ is a fibration and $B$ is a path-connected H-space, is the action of $\pi_1(B)$ on $H^*(F)$ by fiber transport always trivial?
The reason I am asking is that I would like to consider the Serre spectral sequence for fibrations $\Omega^{n+1}Y\to * \to \Omega^nY$ but I'm not sure if I'm allowed to do so.
You are always 'allowed' to consider the Serre speactral sequence for a fibration sequence, but in general, your coefficients will be twisted.
In your case, this twisting will almost always be nontrivial by the following line of thoughts:
Let's consider $\Omega^2X\rightarrow *\rightarrow \Omega X$. The action of the fundamental group $\pi_1(\Omega X)\cong\pi_0(\Omega^2X)$ is given by right multiplication in the H-space $\Omega ^2 X$. (This can been seen directly from the definition of the action.) Hence the action of $\pi_1$ on the homology of the fiber is trivial, if and only if each element acts on the homology by the identity. That's the case if and only if the homology of $\Omega^2 X$ is trivial.