Here's when you'd use each question given by Mrt.
Although which one you use isn't that important and many people may not pick up on the subtle differences between them.
Do you have any classes today?
If you want to find out whether someone has at least one class today.
I would only ask this at the start of the day, since the phrasing focuses on the present and future, yet doesn't exclude the past, so it might be ambiguous as to whether you want to also know about the past. To ask about the future only, I might ask about "the rest of the day" instead of just "today".
This question wouldn't make that much sense if you already know about some of the classes someone has today.
Do you have other classes today?
If you know some of the classes someone has today (in the past, present, or future), or you simply assume they've already had a few classes today (in the past), and you want to know whether they have any more (in the future).
Do you have more classes today?
In this context, this is roughly the same as using "other".
In other contexts, it may have different meanings - if you ask whether someone has "other" burgers, you're asking whether they have other types of burgers. If you ask whether someone has "more" burgers, this may be interpreted the same, but you could also be asking whether there are any burgers at all left.
By emphasising "more", this may also be used to express surprise, in that they've already had a lot of classes, and you're surprised to find out they have even more (you can indeed also make that "even more").
Do you have any [more/other] classes today?
If you already know about "more" classes, and you want to find out whether there are any more in addition to those.
This may be used to express frustration at not being sure whether someone has more classes, e.g. you want to arrange something with them, and they reschedule a few times because they remembered about yet another class.
Do you have another class today?
If you want to find out whether someone has exactly one other class today.
This might make sense to ask close to the end of the day, when you'd expect someone to not have more than one class remaining.
Although I wouldn't expect someone to respond with "no" if they have multiple classes, but rather with either "yes, I have a few" or "actually I have a few".
Do you have (any) further classes today?
A more formal alternative for "more".
Do you have (any) additional classes today?
Do you have an additional class today?
A more formal alternative for "more".
This might be used to refer to classes which are themselves additional, rather than classes that you just haven't been told about yet. For example: you share "core" modules with someone and you ask whether they have any additional classes to find out whether they have classes for any of their electives today.
Do you have another classes today?
This is incorrect, as already mentioned.
An easy way to remember why is "another = an + other" and "an" means "one" (e.g. "I have an apple" means "I have one apple").