Is the veneer actual wood, or is it that stick-on wood-look stuff?
You will definitely need to give it a light sanding. Or you could try a liquid deglosser, although I have never found one that was very effective. Much easier, though, if you find a good one.
You will also need an extra sticky primer, like Kilz-2 or Gripper.
My most important tip: Do sample patches. Lightly sand, degloss in a eparate area if you are trying that. Then do a patch each of a few different primers.
Leave them overnight, then come back in the morning and try to scratch them off. Try your fingernail first. If you can't scratch the primer off with your nail, try scratching gently with something metal. Don't gouge the wood :-). But you want to choose the primer that will really stay on there. If you use something that scratches off easily you will be very disappointed in a year!