If I don't replace it, will it cause any trouble?

 

Your heater won't work as good. Your 'check engine light' might turn on for a system too rich code among others. The computer might even figure it out. I don't really think it would hurt your catalyst enough but it might. Your auto transmission may not upshift normally. It probably won't pass a smog test. Your engine may not reach it's operating temperature in cold environments. That's about all I can think of.

 

And your gas mileage will suffer.

How cold is it running according to the temp gauge?

 

How do you know it's stuck open?

 

And it will result in more carbon buildup within the engine.

For the cost of the repair, it's worth doing and it's not a really difficult job as I remember.

I did the thermostat in my '02 in the driveway mid-winter on a 10 degree day.

 

Yes it will. Lots of reasons stated above, your engine is designed to run at a certain temperature. Running it cooler than that will cause damage in the long-term, as well as cost you quite a bit of fuel economy.