You will get flat soda and an inconsistent brix....i.e. (water to syrup ratio.) The syrup that sits in the line overnight will be warm and will make the soda taste completely flat. The warm syrup will pour faster than the cold syrup. Which changes your brix. The tried and true method is room temp bibs and tubing...running through an ice cooled cold plate immediately prior to dispensing......many years of trial and error.
With post mix...your finished product needs to leave the nozzle below 39 degrees without ice in the glass, or else the soda will taste flat. While the cold plate may be a little inconvenient in the home setting....it is the most effective for a bar gun setup. Any other method will probably cause you much frustration with flat pop.
I install fountain equipment for a living and even the bargun/coldplate is a challenge to pour top quality finished product..compared to the countertop self serve units.
With the bargun....you want the shortest distance from the coldplate to the bargun.....and keep those lined insulated with foam tape....but no matter what...the 1st pour from a bargun after it has sat for a couple hours will be flat.
If you refrigerate syrup boxes, run them 4 or 5 feet and mix it with ambient temp carbonated water......you will have nothing but problems and inferior soda.
Ice it man.....and it will be good. LOL.....hope this helps