I was told that jumping in a western saddle can be very bad for the horse's back. As many western saddles are very straight the horse's back can be EASILY pinched and irritated in jumping.
if youre on a trail and theres like a log or something its ok, but dont make a habit out of it or train a horse to do that. They may not show it yet, but it can very badly hurt them. Dont jump much higher than id say logs or little ditches on a trial. I would not train in an arena over jumps with a western saddle... =/
though like above, you cant alwasy anticipate....one jump wont damage them or anything, neither will little hops over obstacles on a trail. but actually working them over jumps in an arena? Is NOT good