If you still have 30 days to go on your FMT, go to INM and apply for an FM3 Visa. You will need proofs of income, address, your FMT, etc.
If you are not within that time frame, you will have to leave Mexico before the FMT expires or become an illegal alien; not a good idea at all.
If flying out, you must turn in your FMT to INM at the airport before boarding the plane.
Are there problems if you violate immigration laws? You bet. Fines and delays are the least of them and it can get worse. Don't risk it.
RVGRINGO is absolutely correct in that you should definately avoid putting yourself at risk.
That being said, I had an issue a couple of years ago with my car and was a few days late to the border. I was lucky in that I had an English speaking person at immigration that accepted my explanation and even talked to Banercito about car permit which was also late.
I would not recommend this but at least in my case an honest explanation and apology went a long way. I'm not sure how universal but at my preferred crossing point the Mexican authorities are a lot more reasonable than their US counterparts.
you can go to your local imigration office and ask them to issue and extencion..... i am not sure how many weeks or months you can ask for. i had a similar issue and was given an official extencion on my fmt.