If you're sitting at a light and a cop is behind you and you burn rubber when you take off, but you don't break any other laws is that breaking a law?

If it's raining, there's snow on the ground, or many other low traction scenarios, the cop would have to really want to pull you over. Chances are they won't.
On a clear, dry day, depending on where you are, barking your tires is a violation of nuisance laws. Again, most likely not getting pulled over unless you consistently did it.
Now, if you stomped on it and your tires broke loose, you are looking at reckless operation of a motor vehicle. Expect to be stopped.
It mostly comes down to conditions and level of egregiousness... and how bad of a day the cop is having / wanting you to have.
- Oh, I definitely meant broke tires.

As others have said, it depends on the city but probably yes.
For example, in San Mateo, California, it would be an offense under title 11 of the Municipal Code:
11.12.050 EXCESSIVE ACCELERATION OF MOTOR VEHICLES.
It is unlawful for any person operating a motor vehicle within the City to so accelerate the same as to cause audible noise by tire friction on pavement or to cause the tires of said vehicle to leave skid marks upon the pavement, except when such acceleration is reasonably necessary to avoid a collision. Any person violating this section shall be guilty of a public offense.

As others have said it can fall under a variety of offenses. Another offense that refers to this is "Stunting".

It's an "exhibition of speed" and is some form of illegal in nearly all regions.
It's actually more interesting to figure out where it ISN'T illegal.

In New Zealand it would be classed as 'Sustained Loss of Traction' and would be illegal.

Yes.
Source: a cop who used this exact scenario as an example of “letter/spirit” of the law, saying if it was raining and the tires chirped on the crosswalk surface, that wasn’t the same as laying it down on dry pavement.

Same question but instead you only use launch control, so no loss of reaction, and don't go over speed limit.... Although the cop would hear the brup, brup, brup, brup.... Braap... Brrraaaap, and you slow down the rpms to not exceed speed limit....
- If you make your car fart on purpose, hopefully yes.

I heard that it’s called “exhibition”.
- 'exhibition of speed' is what ive heard.

Can’t remember where it happened, but I’ve read about someone being pulled over for pulling away from the traffic lights too quick.

Improper Start is a term used around here by the cops.
- Unsafe start is the term in my area

I wouldn't recommend it. I did this once (didn't see the cop). He just pulled up next to me at the next red light and shook his head in disapproval, but I fully expected to be pulled over. Tbf I'm white and was presenting female at the time, so I imagine it would have gone differently if I was any other demographic.

Depends on where you live and how much of a dick the cop is, ultimately
- That just calculates into whether you get pulled over or get a ticket. He specifically asked if it was breaking a law which has little to do with where the cop is a dick or not. Doesn't technically even depend on the judge.
  
  If a broken law is not enforced does it functionally exist?

It is going to vary from state to state exactly what laws apply, but yes I cop can at minimum pull you over and ticket you for reckless driving. Also, some states have "Display of Speed" laws that include breaking traction. In CA there is a recently passed "Sideshow" law that includes shows of speed, which a police officer might also decide to try to cite you for.