No you do not.
Tensioners and track nuts are seriously overrated.
I have built several SS/fixed and have had zero problems.
My winter fixed has a suicide wheel with a hollow axle with nothing more than those thin locknuts holding the wheel on. (it's an experiment)
Hilly terrain, severe potholes, plenty of temperature swings, severe weather conditions, and salt on the roads.
Plenty of skidding and curb jumping.

Zero slipping or tension problems.

Enjoy