I'm not sure what you mean by affecting hard disk performance or by the internal works of cat, but the thing here is that you are doing very I/O expensive stuff: reading from the disk (the "many files") and writing to the disk (the "one" file).
Combined with that you do this quite often, you are doing a lot of work on your harddisk. If you are doing this on one harddisk, continuously, it will affect performance. Not per se because of
cat, I believe
cat isn't especially an issue here, but because of the intrinsic I/O effect of your functionality.
If you keep on reading and writing from a disk (especially when it is the same), you'd have a noticable performance effect on all other programs that want to read or write. How much is just a question of the speed of the system, if there are any breaks in reading/writing at all, how many disks you are using, etc etc.
So yes there CAN be, but as with all performance questions: you should really TEST if it is the case. That is the only valid metric.