Is the reason given in the denial correct?
Misrepresenting information to the USCIS on a naturalization application will result in a finding that the applicant did not establish the required element of Good Moral Character for naturalization. A misrepresentation need not be material to affect good moral character. If the reason given for denial is correct, it would probably not be worthwhile to file the appeal form.
Does the denial mean that the applicant can lose their LPR status or be deported?
In many cases, the denial is said to be "without prejudice," which means the applicant can reapply after they have reestablished the required period of good moral character- usually 5 years, but 3 years for certain spouses of U.S. citizens. The denial would need to be for a removable offense, which might include a material misrepresentation. Each denial needs to be analyzed as to its particular reasoning.