Should I correct a successfully completed Master's dissertation before publishing online?
You should always correct and polish your work before publishing it. Therefore I would suggest you implement the fixes recommended by your professor, and correct the typos you mentioned. Having done this, you should point out somewhere in the publication that this work was partially adapted from the document you submitted to your institution, stating that you fixed a couple minor things. 

As @darijgrinberg suggests here, you should also remove any official logos and statements that might be included in the document, as to avoid it being confused with an official university publication, let alone copyright breaches.