Are there any drawbacks from using too many include() in a webpage?
I use about 5 include() in my webpages because it makes it much easier for me to make changes.
There is overhead associated with locating and opening files. Whether it is substantial enough to be an issue for your site is something you have to decide for yourself.
If you do start to think that maybe it's becoming an unjustifiable load, you might find some relief of one sort or another in one or another of the caching products out there.