Why can’t the F28 be deiced with one or both engines running?
The Fokker F28 can be de-iced with engines running, but this was not permitted by the Air Ontario manuals.

The Wikipedia article does not list a source for their claim:


  the Fokker F28 aircraft is never supposed to be de-iced while the engines are running because of a risk of toxic fumes entering the cabin of the aircraft


One can however find all details in the final report, which shows an excerpt from a memorandum distributed by Air Ontario's director of maintenance:



This memorandum was not known to all Air Ontario pilots:


  The body of evidence on this point does not support a conclusive
  finding, one way or the other, as to whether Captain Morwood and/or
  First Officer Mills received the Fokker cold-weather operation publication
  or the Mauracher memorandum, which contained, among other
  important information, the crucial proviso, "NEVER: Spray while main
  aircraft engine's are running!!!" (Exhibit 317, p.3). What is clear is that
  a specific pilot bulletin was never disseminated on this point, and there
  is strong doubt, based on the above-mentioned evidence of Air Ontario
  pilots, as to whether pilots Morwood and Mills had received the Fokker
  cold-weather operation publication.
  
  Air Ontario pilots Hansen, Wilcox, Stoger, and Monty Allan testified
  that they were not aware of specific restrictions against spraying the F-28
  for the purpose of de-icing while one engine was running. Only Captain
  Maybury, who had received the Fokker publications, was aware of such
  restrictions. In the absence of specific instruction or a company policy on
  this point, Air Ontario pilots who were questioned in this regard (in
  particular, pilots Hansen, Wilcox, and Allan) testified that they considered
  the practice of de-icing with a main engine running unsafe,
  because of the risk of ingesting glycol into the engine and the danger of
  having a person in a de-icing bucket in close proximity to a running
  engine. However, the evidence of experts in the fields of aircraft ground
  de-icing, aircraft engines, and cold-weather operations indicated that
  aircraft ground de-icing is routinely performed in Europe and the United
  States with engines running. Clearly, an operational matter of such
  importance requires a standard company policy that is made explicitly
  known to all pilots and operational personnel.


(final report, p. 658f., emphasis mine)

This implies that de-icing with engines running is possible, but without
proper guidance or procedures most Air Ontario pilots would advise against
it. The report summarizes its finding on this matter:



The report does not say however, why the original Fokker publication prohibits
de-icing with engines running.