Why did our Lord Jesus not allow the demons and lepers whom He had healed to speak about Him (Mark 1:34/44)?
Our Lord gently revealed a glimpse of His true authority by silencing whatever authority the demon exercised while tormenting the man he had possessed. The demon had recognized Jesus Christ, the Savior, and so expressed his fright acknowledging his own doom which was clearly very near. Jesus Christ's fame subsequently grew rapidly as people questioned the most extraordinary doctrine charged with outstanding unique authority. The Lord would not accept any witness from Satan even if it were true.
As for the leper, through this great miraculous incident, our Lord gives us immense hope in His compassion and willingness to heal us of our many burdens, shame, and suffering associated with our spiritual, physical, and emotional illnesses. The leper was instructed to follow the law he had known and show himself to the priest. This evidences Christ's fulfillment of the law and commandments. His ministry was therefore not an abandonment of the Old Testament; but a disclosure of the radical misinterpretations. Nevertheless, since that time, large crowds followed Christ; perhaps only to witness or experience miracles first-hand. So, it could be said that the Lord had asked the leper not to tell anyone in order to stop the multitude's thronging that might hinder his preaching the good tidings of the kingdom of heaven in other cities.