If you filed on time and paid what you believed was the correct amount, they might be kind and let it go. But don't assume they will.
If you can't file on time, you are supposed to file estimated taxes before the deadline, and to make that payment large enough to cover what you are likely to owe them. If there is excess, you get it back when you file the actual forms. If there is a shortfall, you may be charged fees, essentially interest on the money you still owe them calculated from the submission due date.
If you fail to file anything before the due date, then the fees/interest surcharge is calculated on the entire amount still due; effectively the same as if you had filled an estimated return erroneously claiming you owed nothing.
Note that since the penalty scales with the amount still due, large errors do cost you more than small ones.
And before anyone asks: no, the IRS doesn't pay interest if you submit the forms early and they owe you money. I've sometimes wondered whether they're missing a bet there, and if it would be worth rewarding people to file earlier in order to spread out the work a bit better, but until someone sells them on that idea...