What’s the worst that can happen
The worst that can happen is that you scratch the sensor while attempting to clean it. If you are careful and you use professional cleaning tools this is highly unlikely. Scratching can occur if you get a small piece of grit on the tissue you use to wipe the sensor. Commercial products for sensor cleaning are made in clean rooms and are very unlikely to be contaminated. A typical book of lens tissues is certainly more likely to be contaminated. They are cheaper, but the risk of damage is higher. The sensor covering filters isn’t especially delicate, in fact it’s similar to the coated filters you’d use on a lens, but even so it can be scratched.
If you do scratch the sensor (or more accurately the front of the filter stack that covers the sensor), you’ll get a black line on every image and the filter over the sensor (or in some cases the whole sensor assembly) will have to be replaced. This is a job for the factory service center and will cost several hundred dollars for most cameras. This is clearly not desirable and for older digital SLRs it may not even make economic sense.