I give my production server IP address as App domain in facebook application but it's not saved. It returns error as App domain is not valid. So that I cannot use my facebook connect API.
Can we give IP address as App domain in FB application? If it's not, what is the other way to do?
No, it looks like the app domain can only be a domain and not an ip address.
For example in the app settings page, near the App Domain hover the '?' and see:
Enable auth on domains and subdomain(s)
Also, I found this page: http://developers.facebook.com/docs/appproperties/ it refers to the REST api and so it's not that up to date, but it clearly states there that there are two different properties: base_domain(s) and ip_list.
I suggest that you try to put a normal domain and see what happens.