Typically, a "dual supply" type op-amp stops working if one or the other input gets closer than 1-3V to the positive or negative rail. The voltage range, and how it "stops working", both depend on design. What usually happens is, the input stage goes into cutoff, or input bias is shunted out of the input terminals, and the output saturates to +V or -V. (It's always the same direction, so if the output slammed to +V by pulling +in to -V -- phase reversal -- the amp can get stuck that way!)
It's fairly easy to make an op-amp that works up to one supply, e.g., some JFET amps work up to +V, some bipolar amps (LM358, etc.) work down to -V. The most common method of making an amp that handles both is to simply invert and repeat half the circuit. So the normal input circuit might have PNP transistors (that work down to -V), the inverted section would use NPN transistors (that work up to +V). Somewhere inbetween, the two sections cross over, with the result that input offset voltage changes suddenly (because it's like using two distinct op-amps in parallel, each with its own offset voltage). TLV2371 is a MOSFET amp that does this.
Output stages also vary, for similar reasons. Outputs that can reach the supply rails are more difficult to design, and usually have worse power consumption, higher distortion and so on.
Tim