LAMP has nothing to do with DNS resolution. First, you don't even need the MP part of LAMP unless your site requires PHP and/or MySQL - Apache can work just fine on it's own to host a basic site. But if your DNS is configured correctly, the IP should still resolve even if none of that is running.
However, DNS changes are not instantaneous. When you change a domain's NS record, it can take hours (or more) for it to fully propagate. Even A/MX/etc. records on the same NS server can take a while if they were cached by other DNS servers.
Yes, you're right that you won't need LAMP to host a DNS. However, I'm assuming typical users will probably check to see if their site is resolving by visiting it.