There are 3 owners with money given to them without taxes taken out this year. I put the money into Owner's Distribution Account in the Owner's Equity section of the Balance Sheet. What do I do with these accounts at the end of the year?

For accounting purposes, those distributions are simply dividends, and are usually closed to Retained Earnings after each year end, as with any dividends.

For tax purposes, though, it's important that you separately keep track of the distributions (dates, amounts, recipients). Any simple schedule, such as an Excel spread, will suffice.

Although this separately-tracked info doesn't appear anywhere in your accounting financial statements, it'll play a role both on the S-corp's tax return (Schedule M-2) and in tracking each shareholder's ongoing stock basis (Schedule K-1).