Which is the correct closing remark, Yours sincerely or Your's sincerely?

The correct closing remark is "Yours Sincerely"

Yours is second person possessive pronoun. It replaces "your" + noun.

Yours is the only correct spelling.

Your's is not a correct use of the word. Though apostrophe s is used to show possession, the word "Your's" is incorrect. It should not be used as a closing remark.