Is a teacher sinful if she signs for a salary higher than what she actually receives, because the school administration is obligated to do so by the educational district, knowing that she is only proficient in teaching?

Your signature on documents that prove you receive a higher salary than you actually do is considered a lie and a deception of official authorities, and lying is fundamentally forbidden. However, if you fear that you will be harmed by being dismissed from your job and having no suitable alternative if you refuse to sign, then you may do so to avert harm.