There isn't a risk to putting it out other than the attempt failing, it is just impractical to do so once a large coal vein catches fire. You can read more about the Centralia, PA coal fire on wikipedia which sits on top of anthracite coal veins that have been burning for a little over 50 years and are likely to continue to do so for a few hundred years.
Your proposed idea of "inventing weather modification" is not practical. While we have the technology to seed clouds and produce localized rainfall this still depends on the amount of water vapor in the air. You can bypass evaporating water into the atmosphere, seeding clouds and raining and instead just pump water into the mines and flood them. That will be much more efficient at getting water into the mines.
So why don't they flood the mines? That too is not practical, and if memory serves me right, they did try that in Centralia. The problems are
- Mines open into cave systems
- Water can flow through very small openings
- Coal needs only be hot and in the presence of oxygen to burn.
This means you'll need to keep the coal underwater, isolated from oxygen until you can dissipate all of the geothermal heat mass that has built up in the surrounding rocks from the burning coal. You'll also have to make the mine water tight in order to flood it. You'll also need a way to deal with your flood water evaporating and steam build up.
Faced with the realization that extinguishing these fires is impractical the decisions were made to abandon the towns. There was no real risk in putting the fire out, it just isn't practically possible to do so. The risk in letting the fire continue to burn was abated by evacuating the town permanently. Burning coal does not effect the production of new coal (that takes a long time on geological time scales) -- burning coal is a net reduction in coal mass, not a gain.
As an aside, the other problem with cloud seeding, specifically with the Centralia, PA case, is legal. Pennsylvania farmers were so concerned about the possibility of cloud seeding elswehere in the state depriving them of needed rainfall that in 1968 the PA state legislature passed "WEATHER MODIFICATION - REGULATING RAIN MAKING. Act of Jan. 19, (1968) 1967, P.L. 1024, No. 449" which means you'll need to convince the state to issue a license for your activities, which is not likely to be granted.