Do representational forms of government that limit terms encourage an atmosphere where national identity, fervor and enthusiasm can never really grow?
No. National identity and nationalism have managed to flourish in the United States under representative government. "never really grow" is the sticky part.
But if we slightly alter your question, the answer reverses.
'Do representational forms of government that limit terms encounter a natural disadvantage when trying to grow national identity, fervor and enthusiasm, as compared to forms of government with a more consistent governing body or sovereign?'
Generally speaking, yes seems to be accurate. Of course, this is a broad generalization.
But let us imagine some well-liked sovereign coming to power in some nation. In Scenario 1, he comes to power in a representative state with a term limit of, say, four years. One would imagine that such a well liked sovereign would in those four years foster a respectable degree of increased national identity and nationalism as the population comes to trust in the sovereign's wisdom.
In Scenario 2 the same well like sovereign comes to power in the same nation except that the nation is, in this scenario, a monarchy. Let us say the sovereign rules for 40 years. Without any more information, without the unimaginable and unpredictable complexes of ruling for extended periods of time, we would imagine that this 40 year reign would engender ten times the amount of national identity and nationalism (ignoring, of course, diminishing returns).
Again, a broad generalization. With the unimaginably diverse set of possible circumstances any given nation might face, the complexities of the individual sovereign and his ability to handle his nation's circumstances over a period of time, this conclusion is pure hypothetical approximation. But I think we can conclude that representative governments with term limits are natural less capable of fostering national identity and nationalism than governments such as monarchy.
(nationalism being understood as distinct from patriotism - nationalism, 'my country right or wrong'; patriotism, loving one's country enough to enthusiastically dissent)