How to clean an Apple TV remote

Happened to our play button, thanks toddler... used an alcohol wipe from first aid kit- 70% isopropyl alcohol. Worked within seconds.

Rubbing alcohol dripped onto the buttons, let it fall in the cracks. Boom!

See if these tear-down instructions help:

https://www.ifixit.com/Teardown/Apple+TV+4th+Generation+Teardown/49046

If it is wet, dry it with a dish cloth ASAP.
If it is just gunky from kids or Cheeto/popcorn eating types, I'd wipe with a very lightly moist dish cloth or old T-shirt. Then immediately dry it.
Then finish with a light going over with a microfiber cloth.
I clean my iOS devices with a kitchen counter micro fiber cloth I have set aside just for that.
But, the third party cases around devices get dust, dirt, and hairs in them at some point. Like Otterbox, etc. Then I pull the device, clean out the case. Clean the device. Then use canned air to blast all the elements of the last hairs and specs of dust as I put them back together. I do this final part in a clean place. Like a nice clean and dry kitchen counter or island. Even one hair or fleck of gunk between the screen protector layer and the device will appear if not like a micro surgeon (analog photographer in the darkroom with negatives?) with being clean.
Probably the best way to keep a device clean is, keep it clean!

I am very tidy with tech and keep things separate from dirt and grime and, use protective cases, etc. to prevent disaster.
As a teacher, I see students abuse and treat technology really badly. No cases. No screen protector. Inevitably, they have cracked iOS glass screens (which they keep using!), broken keyboards, and bent laptops. All preventable with cases and screen protectors along with common sense.

I had the exact same problem (kids). The play/pause button was sticky and stayed depressed before slowly rising. I used a "Wolf-Pak Alcohol Prep Pads Isopropyl Alcohol, 70% v/v" alcohol pad and gently rubbed it on the sticky button. I gently rubbed stover the buttons and it started to depress and release the way it is supposed to. I was worried the alcohol would dissolve or erase the finish but so far so good. Worth a try and beats buying a new one if it works for you. Please let me know.

Opening the Apple TV or it's remote will invalidate your warranty. I'd try with a damp cloth (not wet) or a tissue with isopropanol on it.

My problem wasn’t a sticky toddler situation. Moreso a greasy adult situation. I was eating a lunch pack with pepperoni and cheese. The oil from the pepperoni created a grease stain on the remote that wouldn’t wipe off. It prevented me from swiping left/right, up/down 🏾♀😆. So, I got a baby wipe, rubbed all over, dried it.....VOILA!! Working perfectly!