Brennan's in DTD - being a New Orleans restuarant and all - might be doing something. A few years back, we ate there on Mardi Gras eve. They gave us beads to wear, but that was the extent of the celebration.
I have some Mardi Gras merchandise from Disney World, strangely enough... So Disney World obviously does something for it. I can't recall what though. I find this strange since Disneyland is the one with New Orleans Square. So Disneyland might have a pin or something, but I doubt they will do anything big.