Let's say I suspect my friend (or spouse) has Huntington's disease. If I were to take their used tissue out of the garbage can and test it for Huntington's disease, would that be illegal?
There's two things you want to do here.
1) Take the DNA
2) Test it.
With regards to 1), as absurd as this is about to sound, in most US jurisdictions, it depends on the location of the garbage can.
Is the garbage can outside on the curb, waiting to be picked up by the trash collectors? If yes, then the contents are within the public domain, and may be taken by anyone. (This is why everyone always tells you to shred important documents and cut up credit cards, the actual taking of these details in this manner isn't theft.)
If the garbage can is in your house, then you may take the material.
If the garbage can is in their house, you may not take the material, it is theft right up until it's on the curb waiting for the trash collector.
Testing the DNA is trickier. I'm unfamiliar with DNA testing services, but it's probably written in their terms of service that all DNA you provide is your own (and if it isn't, I imagine it soon will be). I can easily see where testing someone else's DNA in this manner can run afoul of several privacy laws.
Regardless, the real danger here isn't criminal liability, it's civil. So let's say you obtain a reliable piece of DNA and send it in. Your friend doesn't have HD. You tell her. She might be upset you took that step, she might be relieved, but regardless, I doubt she'd sue you.
Now suppose she has the disease, which if you're concerned, means there's about a 50% chance she does. (HD is dominant, so if a parent has it, 50% chance child has it).
Suppose you tell her she has it, or withhold that knowledge, you could be sued either way for the distress your actions or inaction cause.
Precedent in these areas is scarce. DNA was only sequenced in 2003. The Genetic Information Non-discrimination Act (GINA) was only passed in 2008. HD itself is very rare.
There are alternatives to covertly testing in this way: You mentioned the case your friend is your spouse. If you have a child together, get the child tested. If the child has HD, so does Mom. If your friend is starting to show symptoms, encourage them to see a doctor.
TL;DR: Criminally, if you do this right, you may escape punishment depending on your jurisdiction. It looks a lot worse in civil, though.
Medical laboratories generally have to obtain consent from the person who's medical material they are testing. Even consumer DNA tests like 23andME require "true and accurate registration" of the person who's DNA is being analyzed.
You could lie to the laboratory and tell them its your DNA, and it might be unlikely that you'd be caught, but you'd certainly be violating the terms of the contract, and it might even be criminal fraud, and therefore illegal.