If you add plugins that are only for the Admin interface, they will have no effect on your regular visitors. If, however, you add plugins that modify the content of your posts in one way or another, the more "filters" they go through will involve some resources to be used.
Additionally, if the plugin utilizes the database in an inefficient way you're going to notice it.
Most plugins are very small and do things very quickly and you shouldn't notice much of a difference. But there are a handful of plugins that are very ellaborate and could potentially slow things down.
If you find yourself in that boat, there are way of storing modified content in the database such that you don't keep applying the modifications every time, but that's something you should look into when you find your site slows to a crawl.