In a sentence does republic need to be capitalized?

The Word "republic" itself does not warrant capitalization if you are simply referring to a state as being a republic, or discussing the idea of a republic in general. For example, this is correct: "America can be described as a constitutional republic."

However, if the Word "republic" is part of a proper noun, such as the name of a country, then, yes, it must be capitalized. An example of this rule would be "The People's Republic of China." It would also be true of a book or movie titled something like "Life in the Republic," where, once again, "republic" is part of a proper noun/name.