iPhone 4S fell into water and blacked out, with only CPU and mainboard burning. Is it possible to export the data from two years ago? In the fall of 2015, my previous iPhone 4S fell into water and blacked out. Afterwards, like now, when charging, the mainboard will get hot, but nothing will show up on the screen and the battery also have no reaction.

Is it possible to fix it? Do you think it's the screen's problem or the battery? If it's the battery, why would the mainboard heat up?

If it's the screen, does it mean after I get a new screen I could see my past data and information on the iPhone?

Or is it possible to save the RAM/CPU, for use in a functioning phone, for me to see the data?
Let’s assume you have valuable data to recover - a reasonable price is $300 to perform enough repairs and labor to get the data back. The repairs would be made by removing the corrosion with an ultrasonic cleaner. See a shop like: http://mendonipadrehab.com/price-list for details.

The videos explain how the device is taken apart, cleaned / and repaired with spare parts enough to get the data back. They literally will recreate enough of the system to power up and transfer data. 

It is very possible, but not certain, that you won't get a phone that works long term, but it’s like someone on life support in an emergency room - life can be supported with the right tools in situations for limited time frames.

If you don’t want to pay - you could invest money in tools, high grade rubbing alcohol, and a toothbrush to clean the corrosion off yourself. Alternatively, select a less skilled shop - but the chances of success for water damage are far less the longer the time between the water and the repair.