I don't think What's My Part: Nose had any prizes mentioned, either (though since none of the contestants guessed the part correctly, then I guess it doesn't matter... though with Cookie Monster being the contestant the prize was probably a cookie, just like his other game show appearances). Did What's My Part: Foot have an announced prize? And did What's My Line? ever award prizes? (if so then it makes sense that What's My Part? and What's My Job? didn't announce the prizes) I also don't think Mystery Guest awarded any prizes.
I've read that Guy Smiley's game show prizes are normally Guy Smiley merchandise, but I don't know of many specific sketches where the prize is relevant to Guy Smiley. The only oens I know fo are Name That Sound!: Grover (where the prize is an autograph picture of Guy Smiley) and Here is Your Life: Sneaker (where, according to Sesame Street Unpaved, the sneaker gets a Guy Smiley ankle bracelett). In most of the game shwos that I know of the prize is either related to the character or the game show subject.
In The Remembering Game the prizes are the items shown on the board (a cookie, which Cookie Monster likes, and a jet plane). In Beat the Time with Cookie Monster the prize is also a cookie. When The Count appeared on Beat the Time he got to pick his prize, and chose another 20 seconds to count. The Anything-in-the-Whole-Wide-World Prize Game also awarded contestants with whatever they wanted, and Oscar chose for Guy Smiley to go away. Oscar and Sally Screamer both chose to trade their prizes (a rubber duckie and a dirty wash cloth) in The Trading Game (Cookie Monster and Bill Smith also traded their prizes in The Remembering Game).
Regarding sketches where the prize was relevant to the game shows themselves, in The Triangle is Right the prize was a trip to the Bermuda Triangle. The prize in The Addition Game was a trip to some palce in california, I cant remember the spelling but I cna only assume that it has to do with math. When Grover apeared on Beat the Time and had to bring in items that contain milk, his prize was a trip to a place that had to do with milk (I can't fully understand what Guy Smiley says at the end of the sketch). And obviously the prize on Pick Your Pet is the pet chosen.
The "Here is Your Life" segments usually awarded chamr bracelettes, parties, and books to remember being on the show. There were other semgnets that had prizes that werenb't relevant to the characters or shows. On What's My Letter? Prairie Dawn won a pet mouse (though the mention of the prize caused her to guess the letter). The Mr. and Mrs. Game gave the contestants a choice between an all-expense paid trip to hawiai (and I think some other things, including a million dollars) or a cookie (but in this sketch the characters weren't major enough to have any recurring interests).