The verb “avoir” can take a direct complement which is a noun without an article to indicate that some scheduled activity takes place. This can work both with “il y a” (“there is”) and with a subject (who the person who performs this activity). Only a few complements are in common use, though this is a productive construction: you can use any activity if the context is clear.

“Il n’y a pas classe” means that a specific class or the whole school does not take place. This can be due to special events or a regular thing.

Il n’y a pas classe à 14h, le professeur est malade.   (There’s no class at 2pm, the teacher is off sick.)
Il n’y a pas classe aujourd’hui à cause de la neige.   (There’s no class today due to snow.)
Il n’y a pas classe aujourd’hui puisque c’est dimanche.   (There’s no class today because it’s Sunday.)

With all of these sentences, you can say “Je n’ai pas classe” if you’re a student (or the teacher). In the positive form, there is no preposition or article before the noun.

J’ai classe à 14h, le professeur est remplacé.
La neige a fondu, il y a classe aujourd’hui.
Réveille-toi ! Les vacances sont finies ! Tu as classe aujourd’hui !

“Il n’y a pas de classe” would have a different meaning. It can have several different meanings, in fact, depending on context: it just means that the class/classroom/lesson does not exist. As a whole sentence, it would be a little weird in a school context. The positive form is “il y a un(e) classe”: this is the usual case where the indefinite article is de in a negative sentence.

Il n’y a pas de [salle de] classe dans ce bâtiment, seulement les bureaux de l’administration.   (There is no classroom in this building, only the admin office.)
Il n’y a jamais de classe après 17h.   (Classes never extend pass 5pm.)
Il y a une classe qui commence à 8h.   (There’s a class that starts at 8am.)
J’ai une classe de sixième cette année, mais je n’ai pas de classe de cinquième.   (I have [= I’m teaching] a first form this year, but I don’t have any second form.)

Some other examples of “avoir” + noun-without-article:

J’ai cours de 8h30 à 16h30 aujourd’hui, et j’ai piscine jusqu’à 18h.   (I have school from 8:30 to 4:30 today, then I’m going swimming until 6.)

Je peux pas, j’ai piscine.   (Literally “I can’t, I’m going to the swimming pool”. Effectively a transparent excuse to decline an invitation, not very polite because it’s obviously false.)

Je rentrerai tard ce soir, j’ai comité directeur.   (I’ll be back home late tonight, I have a management committee meeting. Uncommon, but idiomatic. The long form would be “j’ai une réunion du comité directeur”.)