Riding an untrue wheel on a flat surface should be roughly equivalent to riding a perfectly true wheel on a correspondingly designed bumpy/uneven surface. At least from the point of what forces are acting on the bike and the rider. I.e., if there is a depression in a wheel, an equivalent terrain would be best represented by a BMX rhythm section with its periodical bumps:

It also applies for lateral untrueness: riding such a wheel should feel as rolling over periodic bumps from one side of the wheel.
It means the rider has to adjust to negotiate such "technical terrain" by redistributing own weight to maintain the balance and tire traction, even when riding a completely flat road. As anyone who rode a rhythm section can tell you, not going with its "flow" will quickly throw you out of the balance and possibly off the bike.
Essentially, it is similar to riding with a turned handlebar: it is possible, but you have to constantly adjust your riding style, and your range of steering control becomes asymmetrical and thus limited.
For an untrue wheel, for example braking will not be as effective and predictable because you essentially have periodically pulsing reaction force (both in direction and absolute value), and the friction force will pulse as well. Again, it is as if one is riding a mountain bike with straight wheels in a rock garden - the braking traction varies all the time as the wheels jump from stone to stone. When wheels are in air, the friction is zero.
In the MTB case, having a well tuned suspension (both front and back) helps keeping wheels planted on the ground. I assume the same applies for untrue wheels - having a suspension might hide some of the unevenness, but again it will spend a fraction of available suspension travel, making it less effective when you need it most, e.g. on big bumps.