Required fees stipulated in any contract — including a Lease Agreement — do not stop accruing simply because a court case is filed.
If a contract stipulates fees are due for a specific set of circumstances and the terms of the contract are still in force, those fees can be requested per the terms outlined in the Lease Agreement — before, during or after a case.
However, the above answer does not take into account the nature or language of your specific contract, your state’s laws, or any other circumstance that may be specific to your case. For advice specific to any personal legal questions, Standard Legal always recommends you consult an attorney; you can find a local attorney for FREE at Standard Legal’s Attorney Find page.