Can the Mann-Whitney test be used to analyze two different Likert scale questions answered by the same users? Can the Mann-Whitney test be used to analyze two different Likert scale questions answered by the same users before and after an experiment? For example, question 1 is asked before doing an experiment and goes as follows, do you like reading (scale from 1 to 5)? Question 2 is asked after doing an experiment with the users, and it is as follows: Now, after this experiment, do you like reading (scale from 1 to 5)? I want to show that users did not like reading before the experiment but after the experiment, they started liking reading, and I want to test statistical significance.

The values will be paired (on user), not independent, so no, you would not normally consider the Wilcoxon-Mann-Whitney.
You would use a paired analysis (e.g. paired-t, Wilcoxon signed rank, sign test etc). One important consideration is the precise form of your hypotheses -- especially what specific kinds of alternatives you're interested in making conclusions about (e.g. whether you care about saying something about population means, for example).
If your inclination to Wilcoxon-Mann-Whitney was because you did not want to assume an interval scale on the Likert items, one thing to beware of is the fact that the Wilcoxon signed rank requires you to be able to take pair differences (and only then to be able to rank their absolute values). For the pair-difference operation to make sense (e.g. to call "5"-"3" the same as "4"-"2" and "3"-"1", for example, giving them all the same value $2$, and similarly for the other possible pair differences) you would need at least an interval scale, since otherwise you have no basis to assert the various gaps you give the same value are actually about the same size. This isn't necessarily critical (it's sometimes possible to argue for doing it on some other basis) but if we're going to worry about the ordinal-vs-interval issue (as it seems you were doing implicitly in your question), then it's best to at least be consistent in what we are going to assume.
