To eliminate possible causes of the problem maybe you could replace the U8G2 library with either the SparkFun or Adafruit libraries and see if the problem persists.
Another option that is worth testing is to periodically initialize (.begin) the display as this should initialize all the registers to known values. This is just in case the display has inadvertently been turned off with a command.
I don't think the OLED has a reset that can be activated over I2C/SPI and it looks like the displays reset pin is tied to the D1 reset pin. You could maybe hack the OLED shield so the reset signal goes to one of the spare digital pins instead so you can reset it from code without having to reset the entire D1.
I don't use an OLED & BME280 on the same system but I do use D1/BME280 and for some reason the BME280 plays up and stops sending temperature (all other values are fine) or sends stupid values for everything. Reset does not always work and the safe way to cure the fault is to remove/wait/replace power. I initially though this was a D1 problem but I have once had the same problem with a LoRa32U4 board and a BME280 so now I suspect the BME280 may have intermittent problems and maybe this is causing the display problems you have.
The one device that runs continually with a similar display (Lora 32) that I use as a Lora to WiFi bridge has the display still working fine after about a month of use.