Can I file for citizenship with an open case?

I assume that the "open case" you mentioned refers to state criminal litigation. If that is true, then while you may be eligible to file an application to become a naturalized U.S. citizen, you may expect the USCIS not to approve your application unless and until you are able to supply court-certified documentation showing that the criminal case has been satisfactorily concluded. The intersection of criminal law and immigration law is among the most complex of immigration-related legal matters. In some instances the USCIS will treat a case terminated with pretrial diversion or a similar conclusion the same as if it had been a conviction (i.e. in instances where an admission of the operative facts is a part of the process), and in some instances the USCIS will treat a misdemeanor conviction as a felony conviction. It may be wise for you to coordinate criminal case legal advice from a criminal defense attorney and immigration legal advice from an immigration lawyer in order to facilitate swift conclusion of the open case and in order to maintain eligibility for naturalization.