Are there any pathogenic archaeans?
To date, there isn't a single species that may be considered pathogenic to animals or plants. There are archeon who live in association with animals (in the case of human, they have been found in the gut microbiota aswell as certain skin surfaces) that are mainly methanogens. This could lead to think that archaons could produce "pathologic farting", but that condition has never been described. 

In the case of ascidians and other marine invertebrates, some archeon have been found living in symbiosis, with the microbe aiding the animal by means of nitrogen fixation. One could think that, since they're able to live inside the organisms, they could behave as opportunistic pathogens, but none has been described.

Furthermore, it actually exists at least one known species who could be classified as parasitic, the nanoarchaeota Nanoarchaeum equitans, which has been described as a symbiont of some hyperthermophilic archaeon. Despite that description as symbiont, Nanoarchaeum cannot live by its own, while its companion does. Moreover, its genome contains the typical traits associated to parasitism, with great reduction in genome size and loss of many key pathways and cell functions, including methabolic pathways an even some key components of its transcriptional and translational machinery. Since there is no known function of the Nanoarchaeum itself, it should be considered as a parasite.

However, the lack of evidence may be biased, since few people has used metagenomic approach to look for archeon in clinical samples, and some rare cases may be found in the future.