In some states, it is outright illegal. But some states do allow the removal, under certain circumstances; "if the removal, obliteration, tampering, or alteration is reasonably necessary for the repair". However, that wording can be easily interpreted as meaning not simply for cosmetic reasons because you don't feel like painting around it. And as was mentioned, be sure to clear it with your state first, before it's too late. There is a reason why the manufacturers used the special "rosette" rivets, that are hard to duplicate their installation procedure to look just like factory. A trained eye will be able to spot the difference, even if using the right "looking" rivets. The laws are there to try and protect someone from ending up with a stolen, re-badged car. Back when the laws were written, cars didn't have collectible value like they do now. So now in addition to stolen cars, it's also an issue where a simple VIN swap can transform a solid condition 6 banger car into a rare $200k collectible.
Some negatives of tampering with the VIN the wrong way are:
1) You get a traffic stop by a cop, and he checks your VIN (as they often do). "Uh-oh, that VIN has been tampered with". Your car immediately gets impounded (for all he knows, the car is stolen), and you need to jump through legal hoops in court to hopefully get your car back.
2) You're selling your valuable collectible car, and the buyer notices the VIN has been messed with. "Hmmmm, sure that car is all original".
As a side note. I'm here in Illinois, and I personally know somebody who had his 1st gen Z/28 stolen. The cops did get his car back in one piece, but the only thing that was taken was the VIN tag. They did somehow find the tag, but they would not return it to him, even though he's the rightful owner. He had to get a new VIN issued by the state. Talk about what sucks!

 

The only instance where someone would be concerned with a state issued VIN would be a stock style or concours restoration. They're pretty common here, and nobody worries about them. In fact most people like to see them on hotrods because they know it's been checked by the state and everything is on the up and up.