How to use -F option while Burn bootloader for ATtiny 85 using Arduino IDE
For what I know there is no easy way to pass custom options to avrdude via the Arduino IDE, but Sloeber (to which I'm a contributor) does.

Beware though, the -F option of avrdude command doesn't solve the problem, just forces to skip the device signature verification, but if your Attiny is returning such signature something else is wrong and you'll end up with a failed firmware upload anyway.

If the device was working and suddenly stopped it's more than probably a wiring issue, as already suggested in the comments. The second most probable cause, in my experience, would be a hardware issue: you might have accumulated some static charge and then touched the Attiny or any wire connected to it, destroying one of the ISP pins... 

My bet is on the former: my experience with the latter is you get changing signatures in the form of 0xFFFF00, 0x0000FF and 0xFFFFFF but, luckily for me, this didn't occur so often to accumulate a true knowledge.