They wouldn't. The bias runs in favor of admitting the candidate in spite of formal errors (e.g., misspell address; wrong date on form), and NH is notorious for admitting "vanity" candidates.
But, a fair number of posters here have the impression that a person intending to run has to "get organized" weeks in advance, or else won't appear on a presidential preference primary ballot. IL does have a requirement for 3,000 to 5,000 signatures (I'm not sure that this pertains to candidates running under GOP or DEM banners), but the filing date is well after the filing dates for the four early primary states, or FL.
Tell that to the not one, but two opponents Obama got kicked off the ballot because of inconsequential factual errors.
The bias it toward the letter of the law. Depending on the judge it comes up before, it could be kicked out because she forgot to dot the i in Palin, making the judge think she signed it Sara Pallen.