Candy Cane ball gags in Batman the animated series Christmas With The Joker

The scene as presented doesn't make a lot of sense. It should be easy to spit out a candy cane, and, for whatever reason, the episode doesn't take the time to explain why James, Barbara, and Harvey don't bother to do that. Unfortunately, the non-speculative, in-universe answer is: Nothing is preventing them. The scene just doesn't make sense on its own.

There are two particularly compelling speculative answers, though: one which is I think very well-justified viewing the series as a whole, and one which I think is justified on real-world, common-sensical grounds:

ONE: The Joker of Batman: The Animated Series has a totally unrealistic array of gimmicky props and death traps available to him. In "Christmas with the Joker" we see some giant killer nutcracker dolls and giant killer toy planes that simply don't make sense even in the context of the Laffco factory; over the course of the series, the breadth of whimsical armament the Joker employs is too great to catalog.

It is beyond the scope of this question to investigate where he gets those wonderful toys, but it's always fair to assume that if the Joker wants a nefarious version of an innocuous-looking thing, he can get it. So we can imagine, without violating the established rules of the universe, that these may be specially-designed "gag" candy canes that cannot be spat out because they're covered in a special glue or something.

TWO: We know that it is easy to spit out a candy cane. But we also know that, after they're exposed to saliva for a while, candy canes get sticky. If a candy cane got sticky enough, and it were stuck lengthwise between your lips as seen in this scene, it would be very difficult to remove without the use of your hands (which James, Barbara, and Harvey do not have, since they are tied up in bags).

We also know that the Joker is a sadistic sociopath. So we can imagine, without ascribing special qualities to the candy canes or expecting anyone to act out of character, that the Joker licked those candy canes until they got good and sticky, and then stuck them in the Awful Lawfuls' mouths.

What likely needs to be kept in mind here is two things:

1. This is a kids show.
2. This is an animated kids show.

Ball gags likely wouldn't have gone over well with network big wigs as content suitable for a kids show, even one dealing with some pretty psychotic characters who fire actual guns at the heroes.

Further, animation is a lengthy and expensive process, especially for 1992 when the show (and this episode) originally aired. Replacing it with something more innocuous like a piece of cloth tied around their mouths would require the Joker to untie then re-tie all the gags, which adds to the amount of animation needed and thus increases costs.

One of the reasons later seasons of the show and subsequent spin-offs such as Justice League have a far more simplified animation style and character design was to save on animation costs. Simpler designs that required less time for the animators to draw means less money spent animating each episode.

So, using a candy cane provides a kid friendly "gag" that he can simply remove and replace with very little animation.