What happens when a microcontroller does continuous read openration on an EEPROM IC?
Read operations do not wear out the EEPROM (only write/erase endurance is specified in datasheets). You can do as many reads as you wish.

Reading it very often would suggest some "sub-optimal" software of the microcontroller. My usual approach to data that is used often is to copy that to RAM at startup, because any kind of outside access is painfully slow from the point of view of the MCU.