In Islam, is one is supposed to bury the dead body as soon as possible? In order to become a doctor, the students have to dissect the human body. Is this allowed in Islam or not?

Yes, the Prophet (sws) has directed the Muslims to burry their dead as soon as it is possible for them. When the Almighty has ordained death for a person and decided that he should no longer remain in this world then it is only proper to show haste in submitting to His will by consigning the dead person to the grave as soon as possible. Besides, needless delays in burials cause tension and difficulty for the family of the deceased.

Using a person's dead body after his death whether for educational purposes or for humanitarian purposes (eg organ donation) is alright if the said person in his own lifetime has willed it so. It is not proper to use the dead bodies which are left uncollected from the mortuary, as, I believe, is generally the case. Some missionary doctors should come forward to serve the cause of medical education and donate their bodies for these purposes.