THE PLANE for Europe departs tomorrow, but suddenly the vacationer discovers the all-important passport is missing. Is there time to replace it before the flight takes off?

Probably, especially if the traveler can convince a State Department consular affairs officer that his or her situation is an emergency.

The State Department has established procedures to renew or issue a new passport on a rush basis--overnight or in as little as a few hours, if the issuing passport agency is not swamped. But don't expect special treatment automatically if the need isn't compelling.

Standard passport applications normally take about 10 days. The State Department says it is currently meeting that goal after clearing up a backlog created by an April-May rush. New passport issuances are up about 15 percent this year over 1981 to more than 2 million as of mid-June, with the March-through-June demand setting "an all-time four-month record" due to the stronger dollar overseas, according to the State Department.

A legitimate emergency, says Elee Roeder of the Bureau of Consular Affairs, is, for example, when a family member wants to fly to the bedside of someone who has become sick or injured abroad. But even in a case of simple absent-mindedness, passport offices will "make every effort" to get travelers on their way.

In a crisis, go to the Washington Passport Agency, 1425 K St. NW (open weekdays 8:15 a.m. to 4:30 p.m., phone 783-8170) and explain the problem. Evenings, weekends and holidays, phone a consular affairs duty officer, 655-4000. In either case, applicants will need:

* Evidence of U.S. citizenship, such as a certified copy of the birth certificate, a previous passport or a naturalization certificate.

* Two identical full-face photos, 2 inches by 2 inches, taken within six months, color or black and white. Snapshots and vending-machine photos won't be accepted.

* Current identification, such as a valid driver's license.

* A personal check or money order of $15 for passport fee.

If a passport is lost or stolen abroad, immediately notify the local police and the U.S. Embassy or Consulate. A replacement usually can be obtained from the U.S. Embassy fairly quickly--in a few hours or overnight, depending on the situation. "If you're leaving on the next plane, they'll do their best," says Roeder.

Some form of identification is required. If that, too, has vanished, there may be a delay in issuing a new one. It is helpful to know the passport number so the original application can be verified more rapidly.

For basic application information, phone 783-8200.