See Shulchan Aruch O.C. 442:9 :
חמץ... שייחדו לישיבה וטח אותו בטיט מותר
Chometz which was designated to sit upon (like a chair) and is plastered over, is permissible to keep over Pesach.
The Mishna berura (442:42-44) explains that the chometz being discussed is edible chometz (As opposed to the earlier cases of burnt or spoiled chometz). He explains that although it is forbidden to eat the chometz, it is permissible to keep it and benefit from it.
He specifies that we need two conditions to be fulfilled:
- The chometz is designated to be like a chair.
- Coating it with plaster.
When there is both a definitive designation, as well as an action done to the chometz, it now loses its status as "food" and becomes a regular item. The Mishnah berura discusses whether the whole surface needs to be covered or not.
There is room to equate Play-Doh to the chair and permit keeping it on Pesach, since perhaps adding coloring to the dough is like covering the chair with paint.
The Beis Din Mercazi L'Dayanus rules this way.
See "The Rav Thinking Aloud" pg. 64 and note 58. There, the editor R' D. Holzer explained that Rav J.B. Soloveitchik felt Play-Doh was not edible and therefore permissible.
For others who agree with this approach see here.
On the other hand, as Rav Yisrael Belsky points out, Play-Doh is technically edible. Rav Belsky rules that it is chometz and must be put away and sold for Pesach. There are numerous machmirim who rule this way as well; see here and here for some examples.
(I don't know why they don't equate it to the chometz used as a chair, as mentioned above. Perhaps they distinguishe between covering a surface, and merely coloring the whole chometz וצ"ע.)
All of this is regarding regular Play-Doh which is technically edible, if unpleasant. If you would add cyanide to the Play-Doh before Pesach, then everyone would agree that it is no longer Chometz and can be kept. It shouldn't be played with due to the danger involved.
(You would need to put the cyanide in before Pesach. Once the chometz is prohibited, spoiling it won't help- see Mishna berurah 442:40.)