if your finger was burned, go to the hospital so that they can properly identify your chemical burn.
More likely than not, it's a first degree temperature burn, not a chemical burn. The MSDS sheet for LiIon batteries indicate that flushing any chemically exposed areas with water should be sufficient to mitigate any adverse affects from chemical exposure.
A serious pure temperature burn is another matter, but given that the OP is clearly able to type, it's reasonable to assume his fingers are okay.
Then, take your phone, finger, and hospital bill to Apple and demand a full refund, new phone,
He will likely get one or the other, but not both. Getting a refund OR a new phone will make the OP whole. Getting both is overreaching.
and a reimbursement for your hospital visit. If they refuse, sue. You should not have to go to the hospital because of your iPhone.
He doesn't need to go to the hospital. And if he genuinely did, it would probably be too late by now anyway.
For anyone who may take offense to my suggestion he sue: I'm not the type of person who believes that litigation should always be involved. But at the very least, Apple should be bending over backwards to satisfy the OP.
You're making reactionary assumptions before Apple has even been made aware of the issue, and had an opportunity to bend over backwards (or not). So yeah, people are gonna take issue to that.
OP: Apple definitely needs to do you a solid, but don't start waving subpoenas around until after Apple has had a chance to make it right. Give them an opportunity, if it's a good response for you, then take it. If not, then escalate. But stay calm and don't be confrontational unless it becomes necessary.