Having done quite a lot of reading on the subject I have concluded that ionisers are safe for pets in general. But the safe devices might be ineffective anyway.
I say “in general” because, although the manufacturers tell us how wonderful ionisers are, when you dig around a bit you discover that the benefits of ionisers are unclear and when an ionizer is sufficiently strong to be effective then the machine might produce ozone, and ozone, as I understand it, can be detrimental to the health of people and pets in sufficient quantities.
As a result, the situation is confusing and it is impossible to give a clear black-and-white answer. The way ionisers work is that they create negative ions which attract particles in the air such as dust and dander. As the particles build up the conglomeration becomes heavy and sinks to the floor or it comes to rest on bedding and other furniture where it can be hoovered up or dusted away.
Therefore, ionisers should, in theory, make the air cleaner and reduce the possibility of a person who is allergic to cats suffering from allergy symptoms.
But, as mentioned, if the machine is creating ozone, possibly high levels of the stuff this can aggravate allergies and decrease lung function. Apparently a study found that short-term increases of ozone levels outside might contribute to thousands of deaths annually in the USA.
Clearly, if ozone is produced it could harm a cat depending on where the device is sited and where the cat is allowed to roam. Therefore, it would depend upon the model of ioniser purchased as to whether it could be potentially harmful to a cat or dog.
Also you could argue that if harmful particles in the air sink to the floor or onto bedding as a result of the ions this might be mildly detrimental to a cat because cats spend a lot of the time on the floor or on beds.
My conclusion is that ionisers have dubious benefits to the health of householders and that some models of ioniser may potentially be mildly harmful to cats or dogs for the reason that they emit or create ozone.
If you have better information than me on the subject then I would appreciate a comment below.