Learn Portuguese (Portugal): Dizem que ele vive em Lisboa.

In Portuguese, using a third-person plural verb like dizem without a subject pronoun is a common way to express an impersonal 'they', 'people', or 'rumor has it'. If you included eles, it would sound like you are referring to a specific group of people rather than a general statement.

In English, we can just say 'They say he lives in Lisbon.' Can we drop que in Portuguese?

No, the conjunction que (meaning 'that') is mandatory in Portuguese to connect the two parts of the sentence. You cannot drop it like you often can in English.

Since there is a que, why is the verb vive in the normal present tense (indicative) and not the subjunctive?

Even though sentences with que often trigger the subjunctive, verbs of communication or stating a fact—like dizer (to say)—take the normal indicative tense when they are affirmative. If the sentence were negative (Não dizem que...), it would trigger the subjunctive because it introduces doubt.

Why is it em Lisboa instead of na Lisboa?

Most cities and towns in Portugal do not take a definite article, so we just use the plain preposition em (in). A major exception to remember is Porto, which is a masculine word that requires an article, so you would say no Porto.