I want to know the disadvantages in keeping a 'dedicated' Database Server.

I asked my professor, and he told me tht having a dedicated database server will mean tht there will always be a DNS look up.....which will increase the time required to talk to the database.

Is it always better to have the mysql installation on the web deployment server so tht accessing the DB will ber from the local host??? ?

That depends on how big the project is. For a small project, keeping everything on one server is probably easiest. Your server won't be maxing out its resources and there won't be much benefit from having a separate db server. However, once your project starts to scale up you'll want to have dedicated servers for your databases so they can spend all their processing power on databases rather than sharing it with the application code.

For a basic explanation, check out the 'Scaling' section on this page; it's talking about the web application framework Django, but the discussion of scaling is relevant to most platforms.

Disadvantage i agree is a lookup, doesn't necessarily have to be dns if you use the ip, but once again that might be bad practice.
Advantages would be scalability and most likely replicating the production environment. Issues can exist that you won't realize until production by using separate machines. Assuming you are in school and creating smaller applications, you might not need to worry about it yet. For larger enterprise applications it is ideal to separate out, for academic purposes its not of the highest priority.