In your question, it is unclear if you are using a different database between development and production. Typically, you would have (at least) two copies of the database and would set the connection string to point to the correct environment when you deploy typically by changing the value in the web config. Alternatively, you could set the connection string at runtime using the technique Dave mentions.
In your case, I suspect that you are using integrated authentication and getting cought off-guard by SQL security permissions. When you are testing this in your local environment by debugging it in Visual Studio, it runs under your security credential. Most likely, your user account is an administrator (db owner) in the database and thus you have full permission against the database.
When you deploy this, IIS is running under the NetworkService credential. When you try to access the database, the request fails because NetworkService has not been given rights in the database on each of the tables/views/sprocs/etc.
If you have mixed mode set on your server, you can set the permissions on each database object to a SQL user and set the SQL user and password in your web.config. Alternatively, you can change the user account that the IIS process works under and configure the database to work with that user.