What are the requirements for an aircraft to be allowed to perform fast taxi runs?
Regulation wise, no, there are no regulations which have to be met to conduct a high speed taxi test.  All that's needed is an aircraft with the capability to accelerate to speed and stop as well as a runway long enough to do it on.  If at a towered field, you'll have to get a clearance to do this.

It used to be mechanics would take aircraft freshly out of maintenance out for high speed taxi tests to verify the work was done right.  One such high speed taxi test went ary for a USAF mechanic named George Johnson in 1956 and earned him his first - and successful - solo at night in an F-86!