There were very few theatres in Shakespeare's time, even fewer towards the end of his life when the theatre started to be more rigorously censored (period of James I).
The Globe was circular and open, there may have been some echo in the galleries but not much - it was common of the theatres of the time.
Many plays were performed on makeshift stages by traveling theatres (much of Shakespeare's were written and performed this way with his strolling players).
In his later life Shakespeare performed a number of plays for the crown (James I - rather conversely to point 1.), when he had the 'King's Men' - the plays of this period contain fewer references to Christian religion and more to mythology (especially Greek) so as not to upset the rather strained religious equilibrium of the age. These plays were likely performed in large rooms at the palace or in its grounds giving greater weight to the possibility of an echo. But this would not have been common, or perhaps desired (see below)
Please note that the possibility of an unscripted echo in theatres is discouraged. Even the earliest amphitheatres had specific seating arrangements to reduce echoes. A reverberation, the vibration or tremble caused from sound is often sought but an echo dulls and dissipates the sound and performance.
The occurrence of echoes in modern theatres is reduced using sophisticated architecture and baffles, however techniques have been used throughout the history of theatre. It should be remembered that most Shakespeare plays contain songs, poetical passages and dramatic refrains, echoes would not enhance these.