Wiegand should not be on a twisted pair. If you are using CAT6, do NOT use a twisted pair for your D0/D1. Instead, use a pair of parallel wires such as orange and green or orange and blue (do not use the twisted blue/blue white pair or orange/orange white).
You also need to be wary of power to ensure that your reader has enough voltage at the end of the line to power the reader. Measure this to be sure your reader has the correct voltage (typically reader current is low: in the 100-150 mA range so that should not be a problem but be wary of the voltage. A 12 VDC reader will typically not work at 9VDC...)
Third, you will need to make sure that the wire does not go near any sources of electro-magnetic interference (EMF). Wiegand should use shielded cable and a CAT6 is not shielded. This should not be a major problem as long as the wire does not go too close any motors, transformers, batteries, ballasts, etc... The typical behavior of a reader with unshielded wire is intermittent "non-reads" (a card is presented once and does not read but when presented a second time it reads).
We have MANY customers who have used CAT5/6 like this and if you observe the above you should be ok. But understand that this type of wiring is NOT the correct specified wire for wiegand (which should be a stranded, shielded, untwisted wire of the largest gauge possible-- which is the exact opposite of a CAT6 wire!)