Is it OK for veneers to hurt afterward? Should my new teeth feel a slight bit of pain/sensitivity?

I understand your concern, how long ago were the veneers placed?  If it has only been 24-48 hours, I would give it another day or so to settle.  The soft tissue (gums) may be tender due to inflammation from wearing the temporaries, as well as clean-up of the residual permanent cement after the veneers were inserted.  You could try warm saltwater rinses to see if it helps.  You mentioned on one of the front teeth, the permanent veneer was not placed so I'm assuming your dentist placed back the temporary.  Is the discomfort localized there?  Or is it radiating farther back?  
Temperature sensitivity, ie hot, cold and air are not uncommon for the first 2 weeks.  I tell my patients beyond 2 weeks, there is a potential issue and to call my office to notify me right away.  There are two other possibilities:  the contacts between your teeth (where you would floss) might be very tight with one another - in that case have your dentist take a post-op xray, it might show residual overhanging cement in which case he/she could remove that.  Also how does your bite feel?  Is there a certain area that feels like it's touching first?  Have your dentist check your bite - the way your upper and lower teeth are contacting each other and where they might be colliding when you bite, grind and chew.  There could be an interference or heavy contact that can be producing the discomfort which can be verified with articulating paper.  
Finally I would suggest a custom-made nightguard if your dentist hasn't made one for you already.  This will act as a "cushion" and help absorb forces of clenches and grinding, in addition to helping protect your veneers.  Hope this helps, good luck!  

Hi Alex, 
On the contrary when performing your veneers, it improves the sensibility and your teeth should feel firm and adjusted to your bite.It is not normal that you have sensitivity and extreme pain, you should go quickly to another dentist. In the 10 years of experience and more than five thousand cases attended, this has never happened to us.I hope my answer has helped you.