I've never experienced any performance related issues while using php includes, but like any function, I'm sure if you use it a lot things will start to slow down. Of course in terms of computing a lot is A LOT, I think you'll be fine so long as your 'a lot' isn't some huge amount, I'd venture to say even with as many as 20 you'd be fine, but if that is the case you might have some organizational issues.
But you are talking hundreds of includes for it to be noticeable, Unless your server is an seriously under powered, overloaded piece of crud.
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If you use some unoptimized database queries then the slowdown caused by DB will be much more noticeable comparing to includes. And yes, Chris is right, if your server is not a P1-133 under 1k hits/second the decrease caused by includes will hardly be larger than 1% comparing to the same functionality performed by single solid script.