After looking around the internet, it seems that oils can be used to remove tape residue.
So things like, Olive Oil Coconut Oil would be possibilities for removing the gunk and shouldn't hurt if it gets into cuts.
They should also be good for your hands, especially if they are dry.
outdoors.stackexchange.com/users/8794/Charlie Brumbaugh
Gojo Pumice Hand Cleaner.
If you've worked with grease, grime, or in a coal mine then you know what pumice hand cleaner is, it has silica sand in it to help scour away the gunk.
You use it without water on your dry hands, then after you've loosened all the gunk you wash off the cleaner with soap and water.
I have used this stuff with open cuts on my hands, and it's not bad even when scrubbing the dirt out of your open wounds (small open wounds).
I remember being impressed with how gentle it was as far as stinging is concerned.
Wetnaps and hand sanitizer are the devil on abrasions compared to this stuff.
I highly recommend it.
outdoors.stackexchange.com/users/4148/ShemSeger
Swarfega will remove anything from motor oil to tape gunk
outdoors.stackexchange.com/users/None/
I think the key is to avoid the gunk in the first place, or more accurately the second place.
I like the these tape gloves and the advice at the end.
That advice is you can shave the back of your hands.
The tape gunk comes off bare skin much easier than when it is matted into the hair on your hands.
The second, and more important advice, is to reuse your gloves.
The second time you use them the tape it is much less sticky and you end up with less gunk.
outdoors.stackexchange.com/users/5836/StrongBad
Any detergent-based soap will work best to remove things like oils and adhesives.
Gojo and similar brands are a reasonable choice, and the added pumice lets you scrub a bit harder.
Something like Dawn dish soap will also work, but require a bit more scrubbing.
Ultimately you're down to 1)
The adhesives drying out and falling off 2) mechanical removal (rubbing, scrubbing) and 3) the adhesive being stripped away chemically.
(detergent based soaps)
outdoors.stackexchange.com/users/14346/Monica Apologists Get Out
After using tape gloves for crack climbing, I always have some tape gunk left over on my skin. While it doesn't hurt and goes away a few days of rigorous scrubbing, it doesn't look super great at the office on Monday morning.  How can I remove this gunk left on my skin?  I'm looking for something that's relatively painless, and won't hurt when it gets in cuts on the hand/fingers.  No, switching to rubber gloves is not on the table.
How to painlessly remove gunk left by tape gloves?
outdoors.stackexchange.com
<rock-climbing><climbing><cleaning><crack-climbing><tape-gloves>
outdoors.stackexchange.com/questions/17562
outdoors.stackexchange.com/users/671/Felix
Swarfega, Gojo pumice hand cleanser, and even detergent based soap will remove tape residue. Oils can also be used, including olive oil and coconut oil. Oils, tape gloves, soap, pumice, Swarfega and Gojo are viable solutions.
Pumice, soap, Swarfega and Gojo can work well. Oils and tape gloves can be effective.