Is it permissible for me to pay zakat on behalf of someone without their knowledge due to their refusal to pay it out of ignorance of the ruling, and would this absolve them of the obligation and sin?

Withholding zakat is a grave major sin. No one is permitted to take anything from another's property to pay zakat on their behalf, except for the Imam. Similarly, paying it on someone's behalf without their permission is not valid, because intention is a condition for its performance. Therefore, if this is done, the obligation and the sin are not lifted, and it is obligatory to remind him to pay it.