Remove, pepper juice no.
Deactivate pepper juice effects, yes!
A method that I've personally seen work in an extreme case is :
Mash bananas and rub onto affected areas - typically eyes and area around them.
Apparently ripe bananas work better
Q: Does it work?
A: Oh yes!
This was used on the advice of friends from India.
It worked - story below.
Around the web:
Many sites suggest that washing in milk works
A number of sites say that Aloe Vera in a number of forms works well.
Some say that water works - and many say that it doesn't
I have seen banana work miracles - see below.
The Banana Experience:
Long ago my then-young son decided to play with peppers in a dish.
We told him to leave them alone and that it was dangerous but he persisted.
He was well coated with the juices around and in his eyes before he or we realised.
His skin rapidly reddened in patches and he was in extreme pain.
Crying and yelling accompanied by everything logical we could think of were minimally effective. He was a very unhappy chappie.
We rang friends who had spent several decades in India.
"Bananas" they said. "Ripe ones are best. Squash & apply".
We had none. I headed rapidly for the nearest fruit shop.
Back with bananas maybe 10 minutes later I was confronted by the still screaming, crying red-faced victim.
Just like that. Instant fix.
Whether he was especially sensitive to the banana remedy and whether it would work as well for others, or for all pepper types I know not. But as the advice was based on extensive Indian experience it may indeed be relatively universal.
As a well read child I'd heard that the antidote for Stinging-Nettle burn was to rub Dock leaves onto the affected areas. I live in NZ. We have stinging nettles which are usually far worse than the UK/European variety but they are usually only found in remote areas. Dogs and a few people have been killed by them, but its rare. I once got badly burnt by them, but that's another story.
My wife and I visited Britain and continental Europe in 2003. I and my camera tend to clamber and wander off the beaten path. In England I found that a shortcut off the path in Sherwood Forest took me through groves of knee deep stinging nettles. At Tunbridge castle I tried to walk the olde wall from the Mott to the Bailey. Agh - more nettles (maybe deliberately placed?). In Flanders Fields very few poppies grew. I jumped an irrigation ditch to get closer to some I saw. I landed on all fours in a patch of guess what! And more.
On ALL such occasions in Europe, Docks did indeed grow immediately adjacent and squashing their leaves and rubbing onto the affected areas helped greatly (if not a perfect fix).
Bananas may, just maybe, have worked as well but banana palms are at a substantial premium in the European countryside.
FWIW "Stingoes" spray on anti sting "custom medicine" works VERY well for REAL NZ Sting Nettles. Without it I may have needed to be hospitalised. Interestingly, it claims to be nothing except Aluminium Sulphate.