The sources I know on that topic aren't consistent, it largely depends on the territory (occupied territory or german homeland), the surroundings (urban or rural) and the kind of animal.

From february (in some cases march) 1942 on, Jews in areas controlled by Germany weren't allowed to keep pets at all. The different districts handled this in a different way - in some towns, they put up collection points to bring the pets to, in others the pets had to be put down (most famously Dresden, where Victor Klemperer had to have his 11 year old cat killed). This was only one small step in a line of further discrimination against the Jews, right after the prohibition to use public transportation and only some weeks before they weren't allowed to own electronic devices or bicycles anymore.

In other situations, mostly in rural parts of europe, the situation was different. Farm animals were often taken over by neighbors, cats and dogs just released. I don't have a source for it, but I guess it would have cost too much time and effort to kill those animals as they didn't pose a direct threat to the community.