Why do people omit the definite article?

In the first example, Doctor is being used as the name of the person; the doctor is more of a descriptive phrase. It's short for Doctor <his name>.

tend bar is a set phrase, it's a synonym for being a bartender. It's also similar to the way other people describe their work: a mailman could say I deliver mail, a programmer would say I write code, a garbageman would say I collect garbage, and a composer would say I write music. These are all using the noun to refer to the general concept, rather than any specific item, so no article is needed. You would add an article when you need to be specific, e.g. I write the music in TV commercials.

'Zero article' applies to class nouns, not proper nouns. Titles are class nouns that can be used as proper nouns. 'Doctor' can be used as a title, although you are less likely to hear the nurse say "I'll get Doctor for you."

_{(Adapted from my answer on English Language Learners to a similar question, following a request to migrate that answer to English Language & Usage and discussion on Meta.)}

Let's look at a concrete example before going deeper.

Consider someone who calls himself Cookie Monster.

Saying that he is a cookie monster conveys the idea that there is a group of entities that are each called cookie monster, and he is one of them.

Saying that he is the cookie monster conveys either that the 'group' of entities really has only one member (him), or that he is the most outstanding member of the group.

In each case, the focus is on some kind of classification scheme.

Saying that he is cookie monster says something about him personally - he really enjoys cookies, eat them messily, etc.

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It's a similar case with your bar example. "I tend bar" uses the null article, as distinct from the zero article. (Your doctor example works differently. A regionalism is at play here, using Doctor as a proper noun instead of using a common noun with determiner, 'the doctor'. I don't address this at length here.)

The zero article is the most indefinite article, and the null article is the most definite. Peter Master arranges articles in order from most indefinite to most definite:

zero (Ø1)--some--a--the--null (Ø2)
- Peter Master, "Acquisition of the Zero and Null Articles in English", Issues in Applied Linguistics, 14(1)

Here's an example of the zero article and null article from the same paper:

• Zero article: The boys ate chicken.
• Null article: Mr. Jones was appointed chairman.

The null article example has a similar quality to your "mayor" example.

Note that both zero article and null article refer to something that is absent from the sentence. It can seem a little odd to describe something missing as potentially having two polar opposite possibilities. Masters goes on to say:

The zero and null articles can be readily distinguished by their paraphrasability by either an indefinite or a definite article, respectively

That is, if the sentence retains its sense when you insert an indefinite article, the original had a zero article. And if it retains its sense when you insert a (the) definite article, the original had a null article.

In your example, "I tend bar" uses the word bar in the same manner that "Mr. Jones was appointed chairman" uses the word chairman. It uses the null article.