Which country are you in? Are you planning on adjusting status or doing consular processing? If AoS, what is your visa status?
He can file the I-130's, but it's when you get to the visa application part that he'll need to have sorted it by - and if you're AOS'ing then he'll need to do so asap. He doesn't need to have paid in full, but he must be paying them back in instalments and in good standing with the IRS. And presumably he'll need a joint sponsor if he can't afford to sponsor you on his own.
If he can't get his taxes sorted asap, then you would probably be better off returning to your home country and applying via consular processing, as that will buy him more time to sort his financial situation out.
Good luck.