Payment of taxes for your personal return filed with the IRS always come from your personal account, regardless of how the money was earned.
Sales tax would be paid from your business account, so would corporate taxes, if those apply; but if you're talking about your tax payments to the IRS for your personal income that should be paid from your personal account.
Also, stating the obvious, if you're paying an accountant to handle things you can always ask them for clarification as well. They will have more precise answers.
EDIT Adding on for your last part of the question I missed:
In virtually all cases LLC's are what's called a pass through entity. For these entities, all income in the eyes of the federal government passes directly through the entity to the owners at the end of each year. They are then taxed personally on this net income at their individual tax rate, that's the very abridged version at least. The LLC pays no taxes directly to the federal government related to your income.
Here's a resource if you'd like to learn more about LLC's: http://www.nolo.com/legal-encyclopedia/llc-basics-30163.html