Actually du is not a form of de, but the contraction of de and le. In your example, one would normally say “numéro de téléphone de le directeur”, but to preserve the flow of the speech, the correct form is the contraction du.

For your second example, one would say Papa and not le Papa, so it is “Numéro de téléphone de Papa”.

The same phenomenon occurs with au, which is actually à le.

As Evpok says, de and du are not equivalent, and du is a contraction of de le.

So your question boils down to, when to use an article or not? Which is alas a more subtle question. Usually you would use the definite article when speaking about a definite instance; for example le directeur when speaking about one precise person, the headmaster of your school for example. If however you were speaking about headmasters in général, then you wouldn’t use it. Contrast these two:

La voiture du ("de le") directeur. (The headmaster’s car)

Une voiture de directeur. (A headmaster’s car — meaning, a car worthy of a headmaster. I’m actually not sure there isn’t a difference in English too. Maybe a headmaster car?).

The case of the phone number is similar: usually, one speaks about phone numbers in general, so you get numéro de téléphone. The expression becomes an entity of its own, so that you can use it even when speaking of one particular phone number. Contrast the following:

Un numéro de téléphone. (a phone number)

Le numéro de téléphone du directeur. (The headmaster’s phone number)

Le numéro du téléphone du directeur. (The number of the headmaster’s phone — this is not actually incorrect, but the meaning is slightly different: here you focus on the phone, not the headmaster. If you were a technician repairing it, for instance, you might ask it that way.)

As for papa, as Evpok says, you wouldn’t say le papa when speaking of your father. However, note that you would do it when speaking of someone else’s father, as in:

Le numéro de téléphone de papa. (Dad’s phone number)

Le numéro de téléphone du (again, "de le") papa de Pierre. (Pierre’s dad’s phone number)

Le numéro de téléphone du papa. (The father’s phone number — this isn’t incorrect either, but you’re speaking about the father of an unspecified person rather than your own. For instance, if you were a policeman having just found a lost kid, and you find the father’s number in the kid’s pockets. Although you’d more probably say J’ai trouvé le numéro de téléphone du père instead of using papa in this case.)