The 737 is a very safe plane. It has three fire extinguishing bottles in the wheel well.
On a more serious note, I would get the thing on the ground as soon as possible.
Regardless of the state of the tires, I land anyway. You will figure that out after landing.
All this based on a real experience where on an A300 I burned away a tire during take off, resulting in a immediate , overweight landing and a complete bogey replacement later.
The discussion about it was held in a hotel with a coffee, not in a crippled burning plane in the sky at night.