In the example sentence that you give, "do" is being used as an auxiliary verb for emphasis. In the context of the sentence it is being used to emphasize that a particular situation is real. The word "do" isn't essential to the grammatical correctness of the sentence. It would be possible to say, "Many situations exist..." or "There are situations where..." However, you could not say, "There be many situations where..." or "Many situations be", since "be" is not the correct verb form here. "Be" is generally used with "to", as in "My son wants to be a doctor when he grows up," or with a modal such as: could, should, may, might, must, will, won't, would, etc., as in "I should be there soon." You won't find any examples of "There do be" since "There be" is already incorrect.

Both sentences are incorrect. "There does exist" is correct and "There be" is correct. Languages seem to choose a subject, object, or verb to drop for more efficient conversation. English drops the object for more efficient conversation. Very short sentences like, "Jesus cried" make sense in English because the subject and verb is maintained while the object is dropped. For languages which drop the subject or the verb, sentences missing an object would be unacceptable.