You always need a license from ESRI to use ArcObjects, whether it's a license for a desktop runtime environment or a license to use ArcObjects in a web server application.
Don't confuse licenses with runtimes however. Provided you have purchased an ESRI product that allows you to run ArcObjects, such as ArcGIS Server, you can use ArcObjects in a web server applicatoin without a runtime, depending on your flavor of ArcObjects.