Yes. In fact, the result is basically obvious if you use Czech cohomology on the base.
Serre really had two key insights. First, sheaf cohomology is a pain to compute, but if there is no fundamental group then for fiber bundles the Leray spectral sequence is really just using normal old-fashioned untwisted cohomology. Second, you don't really need to work with fiber bundles -- all you need are Serre fibrations, and those are easy to construct. In particular, you have the standard Serre fibration $\Omega X \rightarrow PX \rightarrow X$, where $\Omega X$ is the loop space of $X$ and $PX$ is the space of paths starting at the basepoint of $X$ and the map $PX \rightarrow X$ is "evaluation at the endpoint". Clearly $PX$ is contractible! An amazing amount of milage can be had from this silly observation!
Serre also really developed many of the key algebraic tricks one needs to work with spectral sequences. For instance, he had the amazing idea that one can work modulo "Serre classes", and thus ignore things like torsion. It's like pretending to localize spaces long before Sullivan and Quillen realized you could do so for real!
Firstly, this is not the correct context to talk about strict fibrations, since the classifying space functor is only defined up to homotopy equivalence. Whilst I believe that there are constructions that make the induced map $BH\rightarrow BG$ into a Serre fibration for a closed subgroup $H\leq G$, these constructions are by no means unique. Rather, one should talk about homotopy fibrations in this context, which are simply the result of turning an arbitrary map into a fibration (say by pulling back from a path space fibration). The point is that this procedure correctly identifies the homotopy type of the homotopy fibre of $Bi$ without asserting the existence of any homotopy lifting property or bundle structure.
Now, with that out the way, let us address the question. Let $G$ be a suitable topological group. Assuming it is compact Lie is certainly more than adequate. Then, as you say, there is a functorially associated classifying space $BG$ satisfying a certain universal property. However, more is true, and in many cases it is useful to take this extra structure into account. Not just the classifying space $BG$ is functorial, but there is a functorially associated $G$-principal fibration
$$G\rightarrow EG\xrightarrow{\pi_G}BG$$
where $EG$ is a contractible free $G$-space. Let us write $\mathcal{E}G$ for this principal bundle. Then a homomorphism $\varphi:G\rightarrow G'$ will induce a morphism of fibre bundles
$$\mathcal{E}\varphi:\mathcal{E}G\rightarrow\mathcal{E}G'.$$
The induced map on fibres will be homotopic to $\varphi$, and the induced map of base spaces is a map $B\varphi:BG\rightarrow BG'$ that classifies $\varphi$.
Note that the sense of this functorality is homotopical, and although the bundle is defined strictly, the spaces $EG$ and $BG$ and maps $B\varphi$ are only defined up to a suitable notion of homotopy.
Now let $i:H\hookrightarrow G$ be a subgroup. Then there is an induced map $\mathcal{E}i:\mathcal{E}H\rightarrow\mathcal{E}G$ as above, and the map of total spaces $Ei:EH\rightarrow EG$ is an $H$-equivariant map which is a non-equivariant homotopy equivalence, since $EH$, $EG$ are both non-equivariantly contractible.
Moreover, since it is a subgroup, $H$ acts freely on $EG$ through this map and induces a map
$$Bi':BH=(EH)/H\rightarrow (EG)/H$$
of quotient spaces, which is easily seen to be a homotopy equivalence using the universal properties. On the other hand, since $H\leq G$, there is an induced map of orbit spaces
$$Bi'':(EG)/H\rightarrow (EG)/G=BG$$
and we easily see that
$$Bi=Bi''\circ Bi'.$$
Now we write
$$Bi'': EG/H\cong (EG\times_GG)/H\cong EG\times_G(G/H)\xrightarrow{\pi_G} BG$$
and use this to identify the homotopy fibre of $Bi''$ as $G/H$. Since $Bi=Bi''\circ Bi'$, and $Bi$ is a homotopy equivalence, it follows that the homotopy fibre of $Bi$ is also $G/H$.
Best Answer
It should be a special case of the Bousfield-Kan spectral sequence for homotopy limits. You can think of it as a "Grothendieck spectral sequence" associated to the "derived functors" of taking fixed points and taking $\pi_0$ (which are, respectively, taking homotopy fixed points / group cohomology and taking homotopy groups).