This has been tested. Two long range planes flew around the earth non-stop with nuclear clocks calibrated to the one in Colorado. One plane going in each direction and refueled in midair.

There was a time gain on one and a loss in the other, but the gain and loss was only in the nano-seconds.

Time will remain constant (it will go on as it has with or without us) but how we mark and measure time will change. A person traveling west to east will see the sun sooner if it is a night flight.
If a plane leaves at 5pm from one time zone flying east with a flight time of one hour to land in the next easterly time zone we would then have to set our watches to 7pm upon arrival making it later than what our body clock is used to. In this sense we are losing time because it is later than we think.
An Airplane leaving at 5pm going west with a one hour flight to the next time zone would arrive at the that time zone at the same time that it departed thereby gaining more time. The one true constant in all of this is that your hotel room will still not be ready when you arrive and there is a 50/50 chance that your luggage is going to arrive later than you do.