on 2005 Volkswagen Beetle

will having the asr/esp light on cause damage to engine

No. But your brakes will be affected. You probably have a poor connection somewhere.
thank you
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1 more answer , 1 more comment
no it seeing it is part of the traction control system ASR stands for Anti-Slip Regulation. It is a early traction system that only prevented the wheels from slipping under acceleration. ESP is a newer traction system that incorporates ASR, BAS, ABS, ETS under 1 control module. ESP prevents the vehicle from going out of control on corners by braking individual tires and cutting the signal from the accelerator pedal. It uses a steering angle sensor, YAW rate sensor, acceleration sensor, and all 4 wheel speed sensors to determine which wheel to brake.
I do have the same problem on my 2005 beetle, the ASR/ESP light will come on for some time then go off, and do not know what to check, can these by diagnose with scan tool like snapon scanner MT1800 ???