Is the sentence "His voice was serious and his manner solemn." grammatically correct? I've seen 'be' dropped in some sentences like this one.
This is a proper example of a "stripping ellipsis." See http://en.wikipedia.org/wiki/Ellipsis_(linguistics)#Stripping
Yes, it's correct. The verb "was" is implicitly carried over to the second clause.
Yes! the verb "was" is implicit in the second clause.
Yes; as others have noted, the verb is implied in the second sentence. The same structure is sometimes achieved using a semicolon:
His voice was serious; his manner, solemn.
This works for other verbs than to be as well:
They seemed to be a different species, and he looked at them and they at him with distrust and hostility. (Fyodor Dostoyevsky, Crime and Punishment)