With a handsaw, one trick is to score through the veneer with a straightedge and sharp knife. Then use a finely-toothed blade, and let the blade do the work rather than trying to force it into the wood --always good practice, but especially in this situation. You may want to experiment with cutting angles, to cut less perpendicularly to the veneer surface.

With power tools, using a sharp, finely-toothed blade and a zero-clearance guide or faceplate can make a huge difference.

With a handsaw, one trick is to score through the veneer with a straightedge and sharp knife. Then use a finely-toothed blade, and let the blade do the work rather than trying to force it into the wood --always good practice, but especially in this situation. You may want to experiment with cutting angles, to cut less perpendicularly to the veneer surface.

With power tools, using a sharp, finely-toothed blade and a zero-clearance guide or faceplate can make a huge difference.

Also, be aware of whether the blade you are using cuts on the push or pull. To minimize damage you want to be pushing the veneer against the substrate rather than pulling away from it...which may affect which side you cut from (unless both sides need to be good, in which case cutting in from both sides and then completing the cut in the middle might help or you might want the equivalent of the zero-clearance solution by sandwiching the cut line between two other pieces.)