To preface, I don’t know if this will mean much of anything in the karting world as my experience it more related to cars. I started my automotive career working for a major tire manufacturer, Bridgestone/Firestone. Although it was on the retail spectrum, we did receive a lot of training on the design and construction on tires.
In the automotive world, there are reasons for directional arrows. Mostly due to the tread pattern molded into the tire, but sometimes related to how the tire is constructed. Given that most automotive tires are expected to last for several thousands of miles it makes more sense than race tires that are expected to last only a few hours or less. The concepts still stand. There are essentially three parts to any tire. The bead (where the tire contacts the rim), the structure (the underlying material that supports the tire) and the carcass (the part that actually comes in contact with the road).
The bead is usually a series of rings of wound up metal rod. The structure can be a mix of metal or other material like polyester woven into a mesh like structure impregnated into a rubber substrate. They then adhere the carcass to that substrate with the the actual rubber that connects with the road surface to get traction.
Whether street or racing tire, the devil is in the details. If the Carcass is a single layer of rubber applied over the entire surface, then there will be a point of overlap (less expensive). Conversely if the rubber is applied by an injection mold, then the rubber will not have an overlap and be completely uniform (more expensive). Most Automotive tires are made this way. Some racing tires are made with the alternate method of overlap.
Now imagine the single layers wrapped around the structure have a overlapping seam, that if runs in the wrong direction are more prevalent to delaminating due to the direction they are run versus the uniformly injected tires. What would you expect to happen? The carcasses comes apart and you are left with the structure (aka cords). There is also a phenomenon called cupping or feathering that happens with tires. Its when the layers of rubber shift and cause an unusual undulation in the surface of rubber that contacts the road. Its more common on automobiles than is with racing due to the shorter lifespan of racing tires. Partly due to tread patterns on road tires versus no tread patters on slicks. It is one reason the dull droning noises above certain speeds happens on road tires. On slicks, however it can lead to uneven pressure of the contact patch meeting the road surface as these little buckle points come in contact. Instead you can get weird vibrations or at the extreme, delamination of the carcass from the tire’s structure.
Like TJ said, Its not very common, but still possible. Given a directional arrow from the manufacturer, I would error on the side of caution. If you need to flip the tires, do so on the rims before swapping sides. We run MG’s at my local club (directional), but I have not tested the opposite rotation theory personally. I have heard stories of delaminating when run backwards. Make your choice and find out for yourself.