It depends a lot on the kind of equipment. Sometimes equipment is designed to have a low impedance output and be connected to a much higher impedance input. Other times equipment is designed so that the output and input impedances are to be matched as closely as possible.
The possible consequences are wide-ranging. Anything from loss of signal strength to overdrive and/or poor low end or high end response. Damage to equipment is also a possible consequence of improper impedance matching at higher power levels.
Speaking as an electronics engineer, the general rule (below radio frequencies), when designing amplifiers and various audio interfaces is try and make outputs low impedance and inputs high impedance. If you connect a high output impedance to a low input impedance you get signal loss and when restoring that signal loss (with gain) you'll amplify the noise floor and this might be annoying.
So, a low output impedance feeding a high input impedance isn't going to attract any measurable signal loss unless you are doing something stupid like feeding a PA output into a microphone or guitar input. Expect the magic smoke if you do that.