There's an example of a smooth but infinite-dimensional fibre bundle $M \to S^3$ with $H^3(M)=0$. It involves some ideas that algori and Somnath Basu have already noted.

The fibre is $\mathbb{P}(H)$, projective infinite-dimensional separable complex Hilbert space. Kuiper's theorem that $U(H)$ is contractible in the operator-norm topology has the well-known consequence that $PU(H)$ is a $K(\mathbb{Z},2)$. Take as clutching function for such a bundle $M\to S^3$ a smooth map $S^2\to PU(H)$ representing a generator of $\pi_2 PU(H) \cong \mathbb{Z}$. Then $M$ is 3-connected by the homotopy exact sequence of the fibration, so has vanishing $H^3$ by Hurewicz.

Since $\pi_2 G =0$ for $G$ a finite-dimensional Lie group (in particular, $PU_n$), this bundle isn't the stabilisation of a finite-dimensional projective vector bundle. To find an example over $S^3$ (or more generally, one that is trivial over the 2-skeleton) with compact smooth fibre $F$, you'll need $\pi_2 Diff(F)\otimes \mathbb{Q} \neq 0$. One can't take $F$ to a surface; I wonder if anything is known about $\pi_2 Diff(\mathbb{CP}^n)$ for $n>1$.

There's an example of a smooth but infinite-dimensional fibre bundle $M \to S^3$ with $H^3(M)=0$. It involves some ideas that algori and Somnath Basu have already noted.

The fibre is $\mathbb{P}(H)$, projective infinite-dimensional separable complex Hilbert space. Kuiper's theorem that $U(H)$ is contractible in the operator-norm topology has the well-known consequence that $PU(H)$ is a $K(\mathbb{Z},2)$. Take as clutching function for such a bundle $M\to S^3$ a smooth map $S^2\to PU(H)$ representing a generator of $\pi_2 PU(H) \cong \mathbb{Z}$. Then $M$ is 3-connected by the homotopy exact sequence of the fibration, so has vanishing $H^3$ by Hurewicz.

Since $\pi_2 G =0$ for $G$ a finite-dimensional Lie group (in particular, $PU_n$), this bundle isn't the stabilisation of a finite-dimensional projective vector bundle. To find an example over $S^3$ (or more generally, one that is trivial over the 2-skeleton) with compact smooth fibre $F$, you'll need $\pi_2 Diff(F)\otimes \mathbb{Q} \neq 0$. One can't take $F$ to a surface; I wonder if anything is known about $\pi_2 Diff(\mathbb{CP}^n)$ for $n>1$.

Added: More precisely, one needs $\pi_2 Diff_0(F)\otimes \mathbb{Q}$ to have non-zero image in $\pi_2 Aut_0(F)\otimes \mathbb{Q}$, where $Aut_0(F)$ is the identity component of the space of self-homotopy equivalences. As shown in a paper noted by Igor Belogradek in his comments above, "Rational type of classifying spaces for fibrations" by Samuel B. Smith, this fails when $F=\mathbb{CP}^n$.