I'm going to expand on what Cujet wrote:
First, what controls ride harshness (mostly) is the stiffness of the sidewall. Generally we are talking about the bead filler - a stiff rubber insert above the bead bundle. What gets sacrificed is steering crispness - what is generally referred to as "Handling".
What controls the speed rating (mostly) is cap plies - the more cap plies, the higher the speed rating. Cap plies are circumferential cords place over the steel belts (which are at about a 15° angle, one angle left, one angle right).
Notice that these 2 things are not mutually exclusive.
So it is possible to have a harsh riding, but low speed rated tire - and a cushy riding, high speed rated tire.
HOWEVER, most vehicles with high speed capability are also desired to handle good. So not only is grip emphasized (and treadwear and rolling resistance sacrificed), but the ride softness is also sacrificed.
By contrast, vehicles on the other end of the spectrum - the ubiquitous 4 door sedan - not only doesn't need high speed rated tires, but ride harshness is not desired, so handling gets sacrificed.
In other words, there is a trend that low speed rated tires ride soft, but don't handle well, and that high speed rated tires handle well, but don't ride softly. But it isn't always true and there are enough exceptions to make this relationship barely true.
Now I want to talk about heat generation.
It is true that the faster one goes, the more heat is generated in the tires. However, the more cap plies, the stiffer the tread and less heat is generated - to the point that even at higher speeds, the same tread compound can be used as on a lower speed rated tire. (Please note: What I am trying to say is that it is only at the extremes where a high speed rated tire needs to have a tread compound that generates less heat.)
Most folks have seen me talk about the technology triangle for tread compounds involving treadwear, traction (especially wet traction), and rolling resistance - that if one property is emphasized the others are compromised. (I'll talk about the exception in a moment.)
Rolling resistance is about heat generation - the technical term is Hysteresis. So tread compounds that wear well and tread compounds that grip well generally generate the same amount of heat. It's the tires with low rolling resistance that don't generate as much heat.
The only place you see truly low rolling resistance tires is OE - the tires that come on new cars from the vehicle assembly plant.
Please note: Tires labeled LRR generally aren't actually low in rolling resistance. The term is used for tires that have lower RR compared to tires with comparable treadwear and traction properties. So it is not uncommon for LRR tires with have RR values HIGHER than some tires not so labeled!
And this brings up the exception I mentioned above. There are things that can be done to a tread compound that will change the 3 way treadwear/traction/RR relationship. However, that kind of change is small compared to how wide the difference between a tire with truly low RR and one that wears extremely well (or grips extremely well)
The most common change in tread compounds is the addition of silica (Yes, a form of sand!) Interestingly, the replacement of carbon black with silica results in improved wet traction - BUT - the way this is usually done is to adjust the mix so the improved RR comes out. (Note: There is an upper limit to how much silica you can add to a tread compound before you adversely affect other properties, such a tear resistance - which affects tread chunking.)
So to answer Char Baby's question: Higher speed rated tires don't necessarily ride worse, but they do often enough that it is almost a rule of thumb.