There are different PoE standards like 802.3af, 802.3at (PoE+) and 802.3bt (announced for 2017). In some articles I found the information that "CAT5e" is the minimum requirment for 802.3at. But "CAT5e" has no requirement for the wire diameter itself. So what diameter is needed to support the different PoE standards in which length without getting problems?
As I know the main problem of transporting power over ethernet cables is temperature rise and resistance.
802.3af defines the max power as 12.95 W and max current as 350 mA over 2 pairs.
802.3at defines the max power as 25 W and max current as 600 mA over 2 pairs.
For both of them the typical wire (24 AWG) supports the temperature rise.
Lower diameters as 26 AWG and 28 AWG increases the resistance, so they are not recommended beside that the temperature rise increases also.
802.3bt will define the max power as 96 W and max current as 1000 mA over 4 pairs. For the moment there are not official documents stating the max temperature rise allowed.
Check this document that goes deeper than my explanation.