As I noted in reply to another question you posted, you need to clarify whether you are asking about Likert items, or Likert scales. See the John-Ubersax link below for more on the distinction between them.
Likert items are discrete (i.e., not continuous), because only a limited number of values is possible, whereas a continuous variable has an infinite number of possible values. Even Likert scales are discrete (assuming there is a finite number of items going into them).
I think a better question is whether one can treat Likert scales (and items) as if they have interval scale properties without dire consequences. The answer is that it depends on who you ask. Some authors believe very strongly that one must not use parametric tests on Likert scales (and certainly not on Likert items). Others appeal to the robustness of parametric tests, and say that it is fine to use them on Likert scales. Others even go so far as to say one can use parametric tests with Likert items (especially if the number of categories is large enough, with large enough often meaning 5 or more). A little Googling will lead you to lots of resources on both sides of this debate. I've appended a couple links to get you started.