Let $\xi : M^3 \to F$ be an orientable circle bundle over a closed orientable surface $F$ of genus $g \geq 2$. I am mostly interested to the case where the bundle $\xi$ is nontrivial. My question is about the mapping class group $\mathrm{MCG} (M) = \pi_0(\mathrm{Diff}_+(M))$. In particular, is this group generated by liftings of Dehn twists and bundle automorphisms?
Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientationpreserving. Every diffeomorphism of $M$ can be isotoped to take fibers to fibers. This is proved using the assumption that the base surface $F$ has genus at least 2, so $M$ contains vertical incompressible tori (unions of fibers) and every incompressible torus can be isotoped to be vertical. If a diffeomorphism that takes fibers to fibers preserves orientations of the fibers then it is isotopic to a product of twists along vertical tori. These are generated by lifts of Dehn twists in $F$ and twists taking each fiber to itself, i.e., bundle automorphisms.
The other possibility is a diffeomorphism taking fibers to fibers but reversing their orientations, hence also reversing orientation of $F$ since the orientation of $M$ is preserved. Such diffeomorphisms certainly exist for the product bundle $M=F\times S^1$, and they also exist for any nontrivial bundle whose Euler class is even. Namely, start with an example for $F\times S^1$ and modify this by removing two vertical solid tori $V_1$ and $V_2$ that are interchanged by the diffeomorphism, then glue $V_1$ and $V_2$ back in by diffeomorphisms of their boundary tori preserving fibers and taking a slope $0$ curve to a slope $n$ curve, using the same $n$ in both cases. This gives a circle bundle with Euler class $2n$. The diffeomorphism of $M(V_1\cup V_2)$ reversing orientations in fiber and base extends over the reglued solid tori since slopes in a torus $S^1\times S^1$ are preserved by a $180$ degree rotation that reflects each $S^1$ factor.
Added a few minutes later: It looks like a similar construction can be made more simply using one vertical solid torus instead of two, where this solid torus projects to a disk neighborhood of a fixed point of an orientationreversing diffeomorphism of $F$. In this case there is no need to assume the Euler class is even.

$\begingroup$ I understand, thank you! Yes, orientability of $M$ and orientationpreserving homeomorphisms are what I was thinking about. $\endgroup$ – Daniele Zuddas Jun 8 '17 at 6:54

1$\begingroup$ The theorem that diffeomorphisms of $M$ can be isotoped to take fibers to fibers is a special case of a theorem of Waldhausen in a twopart paper on graph manifolds in Inventiones 34 (196768). He worked in the PL category but the proof applies also in the smooth category if one uses Cerf's theorem that $\pi_0{\rm Diff}_+(S^3)=0$. Another classical exposition is in Jaco's Lectures on ThreeManifold Topology (1980), Theorem VI.18, which gives the analogous result for Seifert manifolds with nonempty boundary, leaving the closed case as an exercise (using similar methods). $\endgroup$ – Allen Hatcher Jun 9 '17 at 0:36