They started the green card process back in 2010, the case was audited and the department of labor denied it. My company didn't tell me of this fact and just started the process again without my knowledge. I do not want to keep working for this company but am worried that if I leave they will try and make me pay for these legal fees associated with the green card application process. Can they do this?
Just starting the process again without your knowledge should not harm your status in anyway as they are still pursuing your green card -- the audit denial may have been based on a technicality and it was determined that it would be faster to re-apply than to appeal.
If you signed an agreement to reimburse them, then you will need to contact an employment lawyer to see if it is enforceable. You probably know it is important to maintain status if you are changing jobs and get your new case on file before you leave.
PERM fees and expenses must be paid by the employer per Department of Labor regulations. Therefore, you are not liable and any agreement that you pay would not be enforceable.
However, find another employer to file an H1B petition for you before your employment is terminated with the current employer. Consult with your current immigration attorney for case specific advice.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter, nor should it be viewed as establishing an attorney client relationship of any kind.