Is it safe to unclog the kitchen sink while letting the dishwasher run?

TL;DR Just open the dishwasher to pause the cycle, take care of the clog and then resume the cycle.

Unless you don't have an air gap or high hose loop at all, I don't see any problems as long as the dishwasher isn't trying to pump out water into an active clog.

It is a question of backflow prevention.

Under normal circumstances, an air gap (plastic/metal gadget on the corner of the sink with hoses to drain and to dishwasher) or a high hose loop prevents sewage water from backflowing into the dishwasher. However, with a serious sink drain clog, it is quite possible to have one of these two particularly bad situations:

Chemicals or sewage backs up into dishwasher

This really shouldn't happen, as the chemicals are being placed essentially on top of the clog. If the chemicals don't make it through the clog fast enough (before the dishwasher starts to drain) then they won't "jump" to the dishwasher. Even if you half filled the sink with chemicals (not a good idea), the chemicals would not back up into the dishwasher because the high hose loop or air gap should be significantly above the sink drain.

Dishwasher Can't Drain Properly

This is actually what I worry about more, and is an issue with the clog whether you have chemicals working on clearing it or not. If the dishwasher can't drain out the normal pipe then the water has to go somewhere. The usual place would be up the sink drain and into the sink. That is disgusting but normally not a big deal. However, with chemicals in the drain already, that could make for a dangerous mess to clean up.

However, much worse is if the drain is clogged in such a way that the dishwasher drain water can't even get into the sink. In that case, it will back up in the air gap or high hose loop. If you have an air gap then that means dirty water coming out of the air gap and onto your countertop, and possibly breaking the air gap in the process. If you have a high hose loop then it could bust the hose (either at a joint/connection or any weak spot). Or it could just result in dirty water staying inside the dishwasher - it depends on the power of the dishwasher drain pump, the length of the hose, where the clog is located and other factors.