Learn Dutch: Weet jij of het kasteel vandaag dicht is? - Do you know if the castle is closed today?

This is because of het kasteel vandaag dicht is is an indirect question. Indirect questions act like subclauses in Dutch. The word of (if/whether) pushes the conjugated verb (is) all the way to the end of that clause.

Wait, doesn't of mean "or"?

It can! However, when you are asking an indirect yes/no question, of translates to "if" or "whether". You can tell which one it is based on the context and the fact that it pushes the verb to the end of the sentence.

Could I use gesloten instead of dicht here?

Absolutely. Both dicht and gesloten mean "closed". Dicht is very common in everyday, casual conversation, while gesloten sounds a bit more formal, like an official sign on the castle door.

Why does the sentence use jij instead of just je?

Using the full pronoun jij instead of the unstressed je adds emphasis. The speaker is specifically asking "Do you know if the castle is closed?" (perhaps after asking someone else who didn't know).