28F010 flash in a 80C186 System
It's possible to write to the flash, but it can't be written to like normal memory. Write accesses to the 28F010 flash, as with many other flash parts, aren't treated as normal memory accesses. Instead, the values of the data lines during a write are treated as a command. A specific programming algorithm is required, which is explained in Figures 4 and 5 of the 28F010 datasheet.

Writing code to execute this programming algorithm should be possible, so long as there are no other memory accesses being made to the flash memory. In an 80186 system, this would probably mean that the programming algorithm would need to be executed from system RAM, and interrupts would need to be disabled.