(Exec. summary - they're fungi, and aren't animals.)
Eukaryote taxonomy is in a state of flux. What is one day a fungus can be an animal the next. Consider the microsporidia. A friend started his PhD studying them when they were animals. He was, thus a parasitologist. Six months in, they became fungi. He thus became a mycologist, something he's not happy because us mycologists are all slightly grungy vegans.
Part of my work is at the very edges of what is fungal and what isn't. I'd go into more detail, but I won't. It might be very significant, but might not be.
(Most) Fungi are not animals. They are, however, more phylogenetically related to animals than they are plants. Something lost on most people as mycology has been viewed as part of botany, as xbojanx says. And within the fungi, the situation is complex given their polyphetic origin - oomycota and chytridomycota, for example. Oomycota are more related to chromistans such as red algae, and the chytrids show similarities to protozoa. The situation will clarify though, but it takes a lot of grungy vegans talking to other grungy vegan mycologists. The problem with that becomes obvious.
My (just grungy, likes cheese) boss was recently thirty-sixth or seventh author on a big paper aimed at setting consensus on the fungal tree of life. So it should get clearer with time, in case anyone is awake. Incredibly, there are people out there who won't speak to others over arguments what genus this-or-that, the sole representative of which was found by Sir Crispin Blackthorne Fforde Clyster-Pipe's teaboy on his thirteenth expedition to Nuristan and has then sat on a dusty shelf in Kew for the last eighty years should go in to. I'm sorry I just can't muster the enthusiasm let alone the venom for that shit. Rant over.
The situation regarding yeast is simpler though. Yeast can either be basidiomycetes (e.g. Cryptococcus) or ascomycetes. In this specific instance, what people call yeast - baking or brewing yeast. Saccharomyces cerevisiae are ascomycetes.
Bottom line, being ascomycetes, they are definitely fungi.
Other fungi include the Discomycetes (e.g. truffles). They are, of course the fungi that oil up and flex in the mirror with 1.5 kg dumbells.