It depends but generally it will affect your application at the initial load. In scenarios where you have dependency injection, the DLL loading will take longer as the dependency injection tool has to do a bit of work to figure things out.
In regard to explicit dependencies where your application directly references another project or component, performance is hurt on application load but shouldn't affect performance otherwise by its presence. All this of course doesn't account for what's actually in the DLLs as whatever work they do and their size will also increase load time.
There is also the possibility that your application has plug-ins which will decrease performance when they're loaded. When that happens is up to you application however.