Yes! And here's something that backs this up:
Using car in sport mode, turning the car off in this mode, then turning it back on next time I go out, I find I am reverted back to normal mode.
When using car in eco mode and turning the car off whilst set in this mode, the next time I come to use it, eco mode is preserved. It does not revert to normal.
Indeed, if starting the car in eco was not good for the car, it would not preserve it and instead revert to normal.
Using sport from the get go can't be bad for it, it's more just to preserve fuel and maximise efficiency as far as I understand.