Fastest recorded apparent motion of a comet or asteroid seen from Earth (degrees/day)?
This is more of a recent example than a record.
On 2019-07-25, asteroid 2019 OK passed about 65000 km from the Earth at a relative speed of 24.5 km/s.
The Minor Planet Center lists multiple
observations from sites in Italy and Armenia an hour before closest approach.
Using pairs of these (streak endpoints?) by the ISON-Castelgrande observatory, I compute motions of 29.8° per hour at 00:23 UT and 31.7° per hour at 00:26 UT.
The MPC ephemeris service
gives sky motions of 30.6°/hr and 32.5°/hr from site L28 at those times.
If someone had observed the asteroid from (25°S, 85°E) in the Indian Ocean at 01:22 UT, they would have seen it move 76° per hour.

Update: On 2020-08-16, tiny (H=29.8) asteroid 2020 QG passed by at 12.3 km/s, about 9320 km from the Earth's center or 2950 km from the surface.
A NASA news article said this was a new record for close approach distance.
An observer in the right place at 04:09 UT would have seen a maximum sky motion of
$$\mathrm{\frac{12.3~km/s}{2950~km} = 0.0042~radian/s = 860^\circ/hr}$$
From pairs of actual observations reported to MPC,
ZTF Palomar (I41) saw a sky motion of  566°/hr at 10:23 UT,
and ATLAS Mauna Loa (T08) recorded 439°/hr around 11:46 UT.