"Dog Gone People" has a few things working against it: It has basic animation, the gags are kind of predictable, and Hal Smith, try as he might, doesn't really sound like Arthur Q. Bryan's Elmer Fudd.
But you know what? I love the cartoon despite (or perhaps because of) these things. The cartoon is so delightfully corny, with Fudd repeatedly putting his foot in his mouth by treating his boss's dog, Rupert, like a dog instead of a person, saying things like "Oh no, another boo-boo!" and "I goofed again!" There are also silly moments like Fudd eating dog food while Rupert gets to chow down on Fudd's dinner, or Rupert getting drunk on bay rum and then driving drunk, which Fudd is blamed for, or the boss promoting Rupert over Fudd (who's demoted to cleaning the top of flag poles) that are so dumb you can't help but love them.
Hal Smith's decidedly un-Fudd-like voice? Yeah, that's part of the guilty pleasure charm.
The music by Milt Franklyn is a big part of why this cartoon is so fun. He weaves in quite a few catchy vintage songs as part of the background music, such as "Cheerful Little Earful", "Can't We Be Friends", "I Want to Be Happy", "Please Don't Talk About Me When I'm Gone", and of course the favorite "We're in the Money". Referencing old songs was less common in the late '50s/early '60s Looney Tunes shorts, so the practice here gives the cartoon a certain throwback feel that works in its favor.
Overall, if you can check your brain at the door, "Dog Gone People" is one of the better later Looney Tunes shorts.