If one uses a lot of modules, does that slow a script down when it is run?

That is an interesting question and the answer is a definite "yes and no".

Provided you do not go totally overboard with adding modules and do not exhaust your computer's memory so it starts swapping memory to disk, adding modules will have little effect once your program has started running. On the other hand, the loading from disk and subsequent compiling of a lot of modules can take some time, but this is normally a "one time penalty" only, although you will suffer it every time you run the program, hence the benefit of having persistent perl-frameworks such as mod_perl in a web-server environment.

If the compiling takes a long time, perhaps you can have a look at Module::Compile which provides for loading of pre-compiled modules in .pmc format. I have not yet used it, so YMMV.