No it won't Motors forcibly slow down or speed up in response to load all the time. When cornering, the variable speed of the inner and outer wheels just varies the load on each motor.
Lego motors are just cans in a plastic case, with some gears. They're not precision industrial motors with constant-speed controls. There's nothing there that's going to be significantly damaged in the short term by this setup.
One motor might probably draw a few more mA than the other one, and it will run a bit warmer, so if you left the vehicle going round in circles for days, one motor might have a shorter lifespan than the other, but nothing to write home about If you really stalled one, the thermal protection would kick in. I'd love to see Philo comment on the topic, but eh.
If the motors were constant-speed each with their own motor driver, then yes, this would be a valid concern, and the vehicle would try and run in a straight line, like a mechanically coupled axle with no diff. But I've run one-motor-per-wheel, quite a lot.
If it was a problem, this vehicle with one L motor per-wheel wouldn't turn on a dime Especially given it has no weight over the steer axle, so the steer wheels are prone to floating. But it works fine see the video
Sorry for hijacking this thread - I sometimes see a bit of angst in Eurobricks about using multiple motors, it's mostly no drama.