Will having too many running programs slow down my Mac?
No matter what platform you are on, if you have multiple programs all running at once, they are using resources and therefore yes, will slow down your computer. There is a big difference between many applications running, versus actively doing something. 

For example, if you are watching a movie, playing music, and working in Photoshop, all are actively using resources and the system will have to share them.

If you are not actively using something, the a large amount of its resources are given back to the OS, and therefore while still running (and consuming resources), they are not consuming as much. For example - on a Mac you can close all the windows of an application, and most of its resources are released, but it is still consuming 'something'. 

OS X Lion (10.7) takes this a step further with a new feature, called Automatic Termination. According to some of the dev documentation:


  When an application that supports sudden termination goes unused or
  has no open windows, Mac OS X may terminate it behind the scenes. When
  a user wants to use the application again, it usually relaunches
  instantly. Users who want to quit applications manually can still do
  so, but it is no longer necessary.


With Lion, applications that support this Resume/Automatic Termination feature will quit for you behind the scenes to gather resources back for the system if it starts running low. Preview and TextEdit are two apps I have seen this actually happen too. They won't do this if you are working in them, but if you leave TextEdit running and go off to other stuff, eventually it will quit itself for you. 

So, the takeaway is this: Yes, too many processes running will slow down your Mac (and any computer for that matter). This is because resources are limited (obviously more RAM helps you run more stuff, but still a limit somewhere).