Publishing a web app doesn't install new or remove existing software from the server. In general it will simply replace the deployment directory and update the IIS configuration.
In many cases the functionality you need is contained in the NuGet package you reference. However more complex libraries, like Crystal Reports, tend to require software to also be installed. This is generally documented by the package. During development you generally have instructions to install some software on your machine. This same step has to occur on the server as well. Publishing cannot do this. Thus if you are not updating your dependencies then publishing shouldn't have any issues. However if you update your packages and they document that they require newer versions of the third party code then you'll need to manually update your server before publishing.