Are law enforcement able to extract data from an iPhone damaged by juice 4 years ago that hasn't been able to turn on since?

It became damaged by juice (tropicana fruit punch- in other words, toxic chemicals) in early 2014. It wasn't able to power on, and I haven't tried to do anything with it since. However, now I just came across a company in Canada that says it uses forensic techniques that law enforcement agencies typically have to help out the Average Joe like me out. Here in NY I kept hearing the same thing: "you need to have a warrant from a judge to access your own text messages," "nothing can be done," etc. Well, maybe this is a way that could help me out. So detectives and investigators, do you have tools at your disposal that have historically been able to retrieve data from such lost causes?

Yes

Id depends on what the juice did to it, and what data was on it. But generally, yes. People who know what they are doing can pull the chips and see what was saved on them.

That would be quite easy.

The juice would only ruin the power circuit, it won't ruin the memory.

Sure, the memory will be intact. See rock crusher.

They won't even TRY. If there's reason to make the attempt, the Apple store is your best option.
Whoever said you need a warrant to access your own data is either CLUELESS or merely LYING.

Fruit punch is not a toxic chemical.
You do not need a warrant to get into your own computers or phone. They are yours. If somebody else wants to get into it, then they will have to do so the legal way, which could be getting a warrant, or otherwise getting your permission.
These companies will most likely take apart the phone, repair the internals so they are working, and plug them into a computer to get files off of it. They can do this, because you have given them permission.
You can try to do the same thing and just plug it into a computer.