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Gelecek yıl okul başlayacak. means  the school will be started nex year.

Gelecek yıl okula başlayacak. means he will be started to study at school nex year.

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I guess teacher was trying to show the differences between these 2 sentences.

I think the first sentence translates as

The school will start next year. 

(because okul (school) is the subject, and therefore it doesn´t take the dative suffix)

The second sentence has a hidden subject o (he/she), but the object of the sentence is okul, and this takes the dative suffix.

So I think the second sentence translates as

He/she will start school next year.

As to the reason for using a dative suffix on the object of başlamak, maybe si++ can give a deeper reason. Often the logic has been obscured over years of usage. I always think of başlamak in terms of ´to start to do something, to begin to do something´. Although the word ´to´ is only used with translations of verbal nouns.

O yürümeye başlıyor. (He/she is starting to walk.)

Similarly, hoşlanmak that uses the ablative dan/den suffix on the object, I remember by thinking of ´getting enjoyment from something´.