In most cases it is unlawful to pass H-1B fees along to the employee. Fees related to processing a green card may be a different story depending on the facts specific to your situation. Because immigration issues are so fact specific it would be wise to consult an Immigration attorney to discus the details. Feel free to contact my firm. We have a section focusing on assisting with immigration issues.
This response is informational only and is not legal advice. Further, this response does not establish an attorney client relationship.
H-1B fees and Green Card fees are two different questions. Let's discuss H-1B fees first.
The is one government processing fee - sometimes called an Education Fee or an ACWIA Fee (for "American Competitiveness and Workforce Improvement Act" - the name of the law which created the fee) which absolutely must be paid by the employer, and can not be reimbursed by the beneficiary employee. This fee depends on the size of the company - $750 for 25 employees or less, $1,500 for 26 employees or more, and is supposed to go into a fund for the training of U.S. workers.
With regard to all other fees for the H-1B process: to the best of my knowledge, there is no prohibition on employees paying any other H-1B fees, either legal fees or any of the other government processing fees.
HOWEVER: If the employer is ever audited by the Department of Labor, in determining whether the employer is complying with the requirements of the Labor Condition Application concerning prevailing wage, the DOL may subtract anything paid by the beneficiary H-1B employee towards the H-1B process.
For example, if the payroll records indicate that the employee is making $62,000, the LCA promised $60,000, but the employee paid $2,820 towards the H-1B ($2,000 legal fee, $320 filing fee, $500 Fraud Detection/Prevention Fee), the employer will be determined to be $820 short of prevailing wage and so in violation.
With regard to the green card process: there is no prohibition on the beneficiary/employee paying fees for any USCIS process, but again the issue pertains to a Department of Labor process: the beneficiary/employee CAN NOT pay ANYTHING with regard to any PERM Labor Certification process: no legal fees, not advertising fees, no fees whatsoever.
All of this said, even with regard to the fees an employee is allowed to pay, I don't see any basis for requesting reimbursement in the absence of a contract between employee and employer permitting it. An employer, of course, isn't obligated to sponsor a non-U.S. employee for any nonimmigrant visa or green card case.