Well, the logic is sorta complicated (prepare for a Fallout 3 lore dump).
The Republic of Dave's Office of Immigrant Affairs has supposedly "exceeded" their citizenship quota for the year. Therefore, if the player asks Dave (the president, if you couldn't tell) for citizenship, Dave will deny it, giving the Lone Wanderer 24 hours to leave or face "accelerated deportation" (we both know what that means). To speak to Dave in the first place, you have to physically enter the boundaries of the Republic. The same execution threat is given if you insult him, anger him, or refuse to tell him "the secrets of your former nation" upon being granted political asylum within the Republic. If you anger him after becoming a refugee, he only gives you one hour to leave. If you introduce yourself as a tourist, you only have 24 hours before you are executed, and you can't come back after those 24 hours.
TL;DR: You have 24 hours to leave the Republic or face execution.
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