There are at least two types of these.
The first is a "tube" that goes into your regular tire, but is a "solid" foam ring, already made with a "virtual" air pressure, usually available in several gradations from 30psi (maybe even less) up to at least 50psi.
The second is an entire solid foam tire, tread and all; also likely available in the same pressures.
The main issue I have seen reported with both types of tires is the rolling resistance is significantly greater than with a regular pneumatic tire, though I suspect most people reporting that have the lower "pressure" versions.
A secondary issue that's not important to some riders is that there is not going to be the same shock absorption with a non-pneumatic tire, since the air cells in the foam can't as easily and quickly transmit the energy of an impact or bump from one cell to the next as open air could inside an inflated tube/tire, which will easily compress momentarily near the area of impact and push throughout the rest of the tire.
A third issue that is probably very rare that I'm going to guess is easier to cause on the first type is the entire tire/tube rolling off the rim during a sudden hard turn at speed, where the tire is normally scrubbed along it's rolling axis across the road surface with little problem. An underinflated regular tire could do the same thing, but a properly inflated one should not. Since the foam version must be large enough and/or stretch easily to get it onto the rim in it's "already inflated" state, it can't hold the bead of the tire against the rim with as much force as a pneumatic tire would, so side-loading like that has more of a chance to roll it off the rim.
I have a very old set of 700c or 27" (can't read the tire sidewall well enough) ten-speed tires that have foam tubes in them, and even though they are age-hardened they still feel far too soft to me. They're too old and hard for me to get the tire off the rim so I can't see if they have a marking for their brand, model, or "pressure", so I'm not sure if it's because of age or the original design. (They came off a junked bike I received, so I don't know even *how* old they are).
I once thought that I wanted airless tires, but since I started using Slime brand protector strips plus the chunky "tubeless" version of the Slime leak-stopper, well over a year and a half ago, maybe even two years, I have not had a puncture flat of any kind on the protected tires. (they won't stop rim pinch flats, due to placement up on the sidewall area, or punctures from spoke heads and rim defects, for the same reason, but they're great for any kind of tread/lower sidewall puncture). I also have used the thorn-resistant thicker tubes, which can help too, but I think the strips are the biggest thing that has saved my tubes.
I've had all sorts of things stuck in them, and only ONCE did a roofing nail go thru it, and it was a veritable minefield of nails falling off the back of a truck in front of me, with traffic all around and nowhere else for me to go.

I must've hit several of them, but only one made it thru, and it was in exactly the dead center on the rear tire, with my cargo baskets on either side, so the extra weight (which included heavy SLA batteries for a motor test and such) probably helped. But Slime replaced not only the defective strip under their warranty (actually a pair was sent), they also sent me two new thorn resistant pre-slimed tubes! Since then nothing has gotten thru, so it could have been a bad batch, or it could have been Murphy helping guide that nail to the weakspot in the system.
