Will using “Dear Hiring Manager” as a greeting prevent me from getting an interview?
You may have slightly lower chances of success if you use a generic salutation like this, but it should not prevent you from getting invited to job interviews. Furthermore, if many other applicants do the same, your chances will be improved. Nonetheless, you should make an attempt to find the hiring manager's name; if you are completely unable to, then it is better to use a professional salutation like “Dear Hiring Manager” than an informal one like “Hi.”