Once you go to the Consulate and pay your money and take the oath, the Consular Officer sends it on to Washington recommending approval. They don't ask you anything about your finances, or whether you owe any money, or whether you owe any tax, etc.
Eventually, when the CLN is issued, the IRS gets notified, and they of course do take an interest in your finances, but only from the tax perspective. So renouncing might affect any tax breaks you get for student loans, and if so that might affect your final tax return
If you decide definitely to renounce, it might be a good idea to contact whoever administers your loan, and tell them, and ask about whether it affects any payment plans. Apart from that, it's a loan like any other, so you just keep up the payments same as with any other loan.
