To check my created application for different User

I have created a demo application in my asana account in which I have implemented certain API(get a total number of projects). Now the problem is I have to try the same application for the users having different accounts (login through some different organization name) and have to check whether those API will work for his organization or not.

Did you use a Personal Access Token or OAuth to authenticate your application?

Yes I have successfully authenticated my application using OAuth

You will have to authenticate each of their accounts, either via OAuth or by getting a Personal Access Token from them, in order to access their Asana information.