There will be heat - if not from the drive then from the power plant. Conversion of fuel into power is never perfect, and the process will always create a substantial amount of heat.
This conversion, as opposed to the jet exhaust, can be radiated away from the travel path. Some types of fuel generate little to no heat within the engine - a charged ultracapacitor is one example of such "fuel."
Really, humanity doesn't have a usable star drive technology, and we can't say much about what is possible. It's like saying in 1850 that "If someone invents a flying machine, we will clearly see it in the sky by the cloud of coal smoke that its steam engine produces."