Hello.
While I might be proven wrong I believe one of the major constraints in using a metal cutting machine to work on wood would be the speed of the tools and their shape. A too fast turning tool would indeed burn the wood and the wrong shape would shatter it.
Just as in any other subtractive technique the speeds, feeds and tools have to be carefully selected to accomodate for the particular wood workpiece you could be machining. As well as there are differences in machining steel to machining aluminum there are differences in machining solid wood, triplay, MDF or whatever material you are using.
Depending upon the process you can be making the manufacturing procedures can be different. Take for example a simple drilled hole. If you are using solid wood you could make it with the final diameter in one single pass. If you are machining MDF or any other aglomerate you normally have to do it in two passes. One in a diameter smaller than your final and then your final.
There are even procedures used in furniture manufacture that require the workpiece to be drilled to about half its thickness then turn the workpiece upside down and then complete the drill. If not done that way the exit side of the edge of the drill gets chipped.
Taking those comments into account there should be no problem in using a metal router to work on wood. Just make sure you add some suction to keep the work area clean and use respiratory protection if working with aglomerates. While the risk nowadays is a lot less than years ago due to the nature of the aglomerants there are still some other risks like the size of the particules. In any case the "better safe than sorry" rule applies.
I hope this gives you an idea.
Regards.