You bet they do! Just watch this:
The folks at Big Cat Rescue aren’t the first to look into this question, either. In the 1970s, zoologists from the University of Tennessee and the Knoxville Zoological Park gave catnip and smooth rocks sprayed with catnip extract to 33 of the park's big cats to see what they’d do.
The responses were mostly positive. The park’s lions and jaguars reacted most strongly to the catnip, even at very low doses. Both the males and females of these species responded the same way, but reproductive-age animals were more sensitive than either cubs/immature adults or very old animals.
Tigers, cougars, and bobcats, meanwhile, reacted less strongly, and the park's two cheetahs never even approached the catnip or control objects. The researchers noted that the animals that responded to the catnip aren’t ones that would normally encounter it in the wild, since the catnip plant is native to North America and Europe. Except for the cheetahs, the cats that didn’t respond as much would encounter catnip in their natural habitat, and the researchers thought that the difference in the species’ reactions might be because of the plants novelty, or lack thereof.
The animals that did respond to the catnip reacted in much the same way that domestic cats do—sniffing and licking the catnip or sprayed rocks, rubbing their chins and cheeks on it or rolling over and rubbing their body on it. The big difference the researchers found in responses was that, while domestic cats will usually respond to catnip for up to 15 minutes and then take an hour or so of “reset time” before responding again, the big cats’ response can last an hour or more and they show the same response if they lose interest and then return to the catnip just a few minutes later.