「遊び倒す」の意味は、「飲み倒す」や「食べ倒す」と全然違いますか？

Hmm, today, 食べ倒す is used almost exclusively in the sense of 徹底的に食べる. You can use Google to confirm this fact. くいだおれ太郎, a symbol of Osaka, has nothing to do with crimes, either. (飲み倒す is rarer but it's still used in the sense 徹底的に飲む.)

The normal word to refer to dine-and-dash in modern Japanese is 食い逃げ (noun) or 無銭飲食. According to デジタル大辞泉, 食い倒す does mean the same thing as 食い逃げをする, but I believe this meaning is rare or obsolete (I didn't know this meaning).

Lastly, while 遊び倒す is exceptionally common, other combinations are rare. To say 徹底的に食べる casually, 食べ倒す is an option, but 食べまくる, 食べ尽くす or 食べに食べる are much safer. The same for 飲む.

EDIT: 飲み倒す is so uncommon that an ordinary corpus like BCCWJ cannot tell its actual usage. Interestingly, I found an academic article that investigated the actual usage of 飲み倒す. According to this, most people use 飲み倒す in the sense of 飲みまくる. Still, IMHO, while I think I have seen 飲み倒す, 読み倒す and 寝倒す somewhere, they are rare and it's safe to avoid them (even if it means 徹底的に). If you used 飲み倒す in the sense of 飲み逃げ, probably you would not be able to make yourself understood.