Does a slow Time Machine disk affect general Mac performance? Say a Mac Pro tower has OS X + Apps installed on a blazing fast PCIe SSD. It also has Time Machine enabled on a slow 5400rpm mechanical drive. The TM drive is internal and connected at all times.

Will the relative slowness of the TM drive affect general performance when the Mac and its filesystem are being used strenuously? Are internally connected TM drives being being read/written to constantly?
I don't see why one would have anything to do with the other. Drives are not being constantly read from and written to unless something needs to do that. You can play a video game and once it's in RAM it'll stop reading the hard drive in many cases. A large movie on the other hand will constantly read from the hard drive. But even still, Time Machine and your external drive are separate from the other stuff your computer does. While yes, it'll take longer to backup and restore from it, that's the extent of the slowness you'll have from it.