Yes, absolutely they will drown...aka begin to rot very quickly especially in warm weather. Are we talking about potatoes that have matured in the ground or seed potato cuttings?
I have had mature potatoes in the ground show rot within 24 hours of being in standing water. Hence, I always plant potatoes, always, in raised rows. That's in hot weather so they rot fasted than otherwise.
If we are talking about seed potato cuttings, I treat my cuttings with sulphur/ag. lime to prevent rot and again always plant them in raised rows/beds. Even with treatment, and I might add even with raised rows, you can get some rot after about 72 hours...much quicker without treatment and/or raised rows.