AP mage isn't quite a kill lane. Mage supports are certain mages who offer enough utility through damage, mainly zoning power, to try and burst down the ADC or pull their ADC ahead in CS. Their counters are that they tend to be squishy, slow, and telegraph skill shot oriented. Extremely susceptible to ganks as they naturally shove the wave due to spell collateral. Also a high CC engage, Leona or Thresh, can easily bring one down as now they are a higher threat than the lower damage ADC early game. So unless you're Soraka, any damage inflicted on them can't really be sustained as they don't have Coin usually or Relic for the heal. Spellthief's offers nothing, and the mage supports bot don't have natural sustain.
A kill lane is one with a champion not meant to be in a support role who just has high base damages and offers really nothing but extreme kill potential.
Examples: Pantheon, Darius, Zed, Lee Sin.
Their punishment? Lack of gold. Kill lanes have one objective. Hard snowball. Anything short of that and they are almost instantly a loss. That or end the game by mid game. Once mid game and end game roll out, the support kill role is underfed and becomes half useless. Champs like Panth and Lee Sin already fall off late game, imagine them with a fraction of the gold. However, in lane, they have a huge advantage in that early game isn't really reliant on gold, especially before first back. Then it's completely even, so Two high damage beats one high damage and a medium damage. Versus a kill lane, play safe, they will also naturally shove, and wait for ganks. The counter to a kill lane is not to beat them or even out CS them. It's to make sure that kill support doesn't get gold, or at least as little as possible.