Does driving at high RPMs (3000-3500) in a manual transmission car cause any extra wear and tear to any mechanical parts? It obviously hurts gas mileage, but does it hurt anything else?
It all depends on your definition of high.
In my car, the red line is at 7500 rpm, and that indicates that driving with the revs over this line for anything other than brief periods is expected to cause damage, either through overheating, increased wear, increased loading on bearings, lack of sufficient oil/fluid flow etc.
When driving I have to keep my revs below 3000 until the engine and turbo are at the right temperature, then I typically keep my revs between 3000 and 5000, as that is the best range for it, but you need to look at what your car is designed for.
The maintenance intervals are defined by mileage, so as long as you follow the guidance in your manual on revs your engine should be fine. As Krom commented, your manual will often give a 'green' zone for engine revs - which will indicate where your engine should be running.
Incidentally, running at too low rpm can also cause problems, which is why the green zone may be 1500-3000 or 2000 to 3500 etc.
Within a certain range, of course more RPMs mean more wear. Especially if your maintenance is based on time or miles.
Consider a bearing that has a lifetime of 1,000,000 revolutions. If you drive at 5,000 RPM, that bearing is going to use up its lifetime twice as fast as if you were driving at 2,500 RPM.
On the other hand, "lugging" an engine at too low an RPM can be harmful, as well.
If your prime directive is mechanical lifetime, shoot for lower RPM within a reasonable range (i.e. not lugging).
The absolute most stress on an engine occurs at high RPMs. The forces acting upon the rotating assembly are exponentially increased with RPM. That also increases the magnification of any oiling inconsistencies (as small as they may be nowadays), which can be deadly to engines.
Increasing power (cylinder pressure) has a much smaller impact on engine stress, relative to RPM induced stress.
The lower RPMs you can stay within, the less stress your engine will endure. This is also why lightweight and balanced rotating assemblies contribute to much longer engine life at higher RPMs.