Can 230V 3phase motor running on 208V 3phase?

To properly answer this question, we would need to have the actual motor ratings and operating load conditions, I.e.; nameplate voltage, frequency, amperes & service factor. Actual operating conditions would be; operating voltage, operating amperage & type loading (constant, cyclic/constant etc.). Generally, a motor with a nameplate rating of 3ph 230 volts can operate as low as 3ph 207 (+/-10%) system volts. However, if the operating conditions are a heavy load at or near nameplate full load rating, operation at 3-phase 208 volts could result in higher operating amperes, resulting in higher operating temperature. Higher operating temperature will result in a shorter life expectancy.

Even though the system voltage of 3 phase 208 is within the -10% rating of the three phase 230 v motor, the 208 is allowed to vary to -10% or 187 volts -- and the motor will not work at that level. If you can't get a 208 rated motor (or one which has multiple voltage ratings like 208-230, which are made) -- then you're better off looking to insert transformation to get back up to 230 volts; this could be a 2-winding transformer of 208-230 volts or else you could get three (3) buck-boost transformers (2 winding units configured as auto-transformers) to get you there.

This motor was built to be used on 240V system. Torque is current X flux. Flux is reduced to 208/240=200/230=86.7%. The load torque demand is the same at the same speed, so the current will adjust itself at 115% of what was planned to be pulled by the load originally. If the motor has a Service factor of 1.15, everything is fine, which is usually the case.
On the other end, reducing the voltage drop might solve the problem if the following assumption is considered acceptable.

Normally, designer in a correct mind specify a motor to work at around 90% or less of the motor capacity. Under this assumption, the motor could be overloaded by 90%/86.7%=1.038. Then, we can expect the motor to be right at nominal current if the voltage drop in the cable is very small, and the motor is fed at 3phase 208v. Then the current would be 90%/(208/230)=99.5% I rated.

Another concern is the motor appears to be for a 50Hz three phase power supply, but the power supply on site could be at 60Hz 208v, so we must take that into consideration, too. At 60Hz, the motor will run at 20% higher speed than it would at 50Hz, so it may require a further 20% increase in voltage.