I was wondering around in the middle of Winterhold and Dawnstar and I saw argonian fast asleep.

I burnt him to death and realized that he was named 'Deekus'

Then I got worried as I had already saved my game and later found out that he was a DB contract!

I'm planning to join the DB soon, will there be any problems?

I don't think so. I tested this with the "Mourning Never Comes" and Nazir's missions. It might bug out later, but it hasn't so far. All it did with "Mourning Never Comes" was give some new dialogue options for Muiri and it prevents you from being able to marry her. That seems to be it.