Are there any serious dangers to using manual tools?
Anything that cuts wood will cut flesh.

While hand tools are in general somewhat safer (since you'll stop when you cut yourself) it's quite possible to damage yourself to a hospital-visit-required extent (before you stop) through mis-use of hand tools, especially if you think they are magically safe since there's no motor.

I have a guillotine miter trimmer that will take your fingers off if given half a chance. I consider it rather more dangerous than my circular saw, as a matter of fact - if picked up carelessly the massive razor-sharp cutting head will move on its own and if anything is in the way it will be cut either deeply or off.

A rather more common tool that can hurt you badly if you are not careful is the humble chisel. And that hand saw can cut tendons before you stop if you get in its way. For the most part, dull tools are actually more dangerous than sharp ones (of tools that should be sharp) since a dull tool requires more force to use, so when it goes awry it goes awry with more force behind it. 

A good deal of situational awareness encompassing "how this could go wrong and where the tool will go in that case" is a pre-requisite for safe tool use, powered or not. As are details like actually using safety glasses when you use a hammer (chips from the object being struck can be flung into your eyes with no need for power to be involved.)