You're really asking two questions here:
1) Do road tires lose air more quickly?
And 2) Do tubeless tires lose air more quickly?
First off, let's talk about the different ways that tires can (and do) lose pressure.
Obviously, they can lose pressure through a poor seal, either on the valve or where the tire seats to the rim on tubeless tires.
Tires also lose pressure through permeation. This essentially means that air molecules escape between the molecules that make up the casing (i.e., tube or tubeless tire). This is true of all tires, not just bicycle tires.
And of course, there is always the possibility of a slow leak caused by a puncture.
No matter what the reason for air loss, a tire at a higher pressure will lose air more quickly than a tire at a lower pressure. Thus, if we assume the same type of tire, materials, etc: road tires lose air more quickly than cross tires, and cross tires lose air more quickly than mountain bike tires.
Additionally, air loss is also more noticeable on smaller tires because any air that escapes represents a larger percentage of the total volume of air. So, if you have a 26x2 mountain bike tire and a 700x23c and inflate them both to the same pressure, they will initially lose the same amount of air at the same rate. However, since the air lost in the road tire represents a larger percentage of the overall volume, the pressure will drop more quickly. And as the pressure drops, the road tire will lose air more slowly, but the pressure will still drop more quickly than the mountain bike tire due to its significantly greater size.
Air loss is also exacerbated by thinner walled casings. This means that (again, assuming the same materials) lighter and thinner tubes will lose air more quickly than heavier and thicker tubes.
Now, I don't have a lot of experience with tubeless, but given that caveat...
Tubeless tires should in theory lose air less quickly than tubes due to the thickness of the casing and the fact that people typically run them at lower pressures. I have heavily stressed "in theory" here because it is certainly possible that tubeless tires are more permeable than tubes and because of this, some balance is achieved.