Its apparent size and brightness in the sky would fluctuate a couple to a few times every night.
GIF is (...still?) broken, will try to fix later
The minor planet has a rotational period of about 4 hours and is actually about as large as our moon: 2,100 × 1,680 × 1,074 km. (On it's long axis it's actually larger than the Moon!) At the moon's orbital distance, the visual changes would be most apparent to anyone on Earth.
Haumea has an albedo of 0.6-0.8 (whereas the Moon has an albedo of 0.12) due to its icy composition. It would be significantly brighter than the Moon, if it weren't for the fact that it would sublimate like the mother of all comets being so close to the Sun.
If you were to replace the volatile material of Haumea with the Moon's regolith and pockmark it with thousands of craters, it would effectively look like our Moon in the night sky (though just a bit smaller on average), stretching and squashing itself throughout the night. Any visible features would give away to any onlooker that it isn't actually changing shape, but is oddly shaped and rotating.