This depends on the university. Each university will have its own approach to handle the delay of applications, start terms, etc. If you have been admitted, but your start term is delayed, the university will most likely not ask you to go through the admission process again; however, you may be required to submit new financial documents in order to receive a new I-20 immigration document.
If you have not been admitted, but your application was submitted, the university may require a new application for the upcoming semester if they have delayed the term for which you have applied, but keep in mind that you will already have the documents prepared to resubmit or the university may keep your submitted documents on file. Since this varies, it is recommended to ask each university what their process would be for this situation.