I am trying to understand how weak convergence works in the theorem of existence of weak solution from Evans (Theorem 3, chapter 7.1), in particular how pass from (30) to (31). Here the theorem with proof 
Does (31) follows from (30)? Is it necessary, passing from (30), to have (31) or can I have (31) only with weak convergence of sequence?
(31) does follow from (30). If the two sides of (31) are different on a positive measure set of times, then possibly restricting to a smaller set you can assume without loss of generality that $$ \langle \mathbf{u}',v\rangle + B[\mathbf{u},v;t] > \langle\mathbf{f},v\rangle$$ on a positive measure set of times. Then you should be able to cook up a test function $v$ (which depends on both space, and more crucially, time) to force the same inequality in (30), leading to contradiction. Something like the Lebesgue differentiation theorem should get you started.
(30), or some analogue, is required to conclude a pointwise a.e. equality like (31). It is well known that weak convergence implies nothing whatsoever about pointwise convergence: there are weakly convergent sequences that do not converge pointwise anywhere, e.g. the typewriter sequence. This is typical in PDE: if you want to generate a solution with nice properties by using a sequence that converges weakly to it, it is typical that at some point the weak convergence needs to be upgraded to some stronger notion of convergence by using the properties of the PDE.