DS3231 RTC module and battery problem
If i keep using the CR2032 battery for long run 24/7, how long will battery last? is there a chance of explosion?


Yes. Recharging a non-rechargeable battery is dangerous. It should be avoided at all costs.


  is there any other way than modifying the module it self?


You have to prevent the power getting to the battery. That power is fed through a diode (hence 4.3V, which is 5V minus 0.7v for a silicon diode forward voltage).  Without knowing what the module is, I can't tell you if there is any other way that modifying it - however I would guess that no there isn't. You need to stop that power getting to the battery, and that can only be done at some point between where the power comes in to the module and the battery itself - and the simplest method is to remove the diode.

The purpose of that diode is to prevent power from the battery from trying to power the rest of your circuit when you remove the main power. Removing the diode will have no adverse affect.  You just won't be able to charge a rechargeable battery in future.