I have no need to inadvertently save money.
Your aim should be to owe nothing, but also to receive nothing back. If you are getting money back it means you payed too much in the first place.
No matter what you use your refund on, you would be better off not overpaying and instead having the money in each paycheck. If you put your refund into savings, you would be better off putting the extra money into an interest earning account each month. If you use it to pay off debt, you could have used the money to not accrue the debt(and pay ridiculous interest charges) in the first place. If it's used to pay outstanding debt you could have been paying it each month. If you put it towards a big purchase, you could have put the money into savings each month and earned interest on it instead.
I see no advantage to overpaying your taxes.