Yes, you can start a sentence with 'Or' if you are writing in an informal style. In a formal piece of writing, business letter, or official document, it would be more appropriate to begin with 'Alternatively, ..' for example. It's the same with words like 'But' and 'And'. In formal writing you'd use 'However' and 'Furthermore', but in everyday informal writing it's fine to start sentences with these conjunctions. My main objection to the correction you've quoted here is that fact that 'Then I'll go to the beach' doesn't mean the same as 'Or I'll go to the beach'.

In written text it isn't grammatically correct to start a sentence with 'or'. In spoken text (i.e. writing a story and your character speaks) then something like "Should I get a dog? Or should I get a cat?" is acceptable. The word 'or' is, in it's simplest terms, a word that connects two sentences together. It replaces the period.