May a Kohen enter into a room that contains the cremated remains of a loved one?
Unfortunately, cremating the dead has become all too common amongst people including irreligious Jews both in Israel and around the world. Doing so is severely forbidden according to Jewish law and desecrates the honor of the deceased, and raises the following questions vis-a-vis a Kohen. The ash of the loved who was cremated is commonly kept for sentimental purposes in a jar in the home of the children or other close relatives of the deceased. The question arises as to whether a Kohen may touch, or enter into a room that contains such ash. This question becomes common when visiting irreligious Jews for Shlichus purposes, in which many a times one may encounter such a scenario.

Does the ash of a corpse give off Tuma?[1]
The ash of a corpse does not give off impurity and hence there is no prohibition for a Kohen to

May a Kohen enter into a room that contains the cremated remains of a person?
Yes. This applies even if the ashes belong to a Jew. Cremated bodies are burnt at temperatures between 1400-1800 Fahrenheit for about 2-2.5 hours. The bones are reduced to calcified matter which are later crushed and processed, being reduced to ash like particles. Thus, in a jar containing the cremated remains of a person there are no remaining body particles that pose a Halachic issue for a Kohen regarding the laws of impurity.

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[1] Rambam Tumas Meis 3/10; Chachomim in Mishneh Ohalos 2/2

Other opinions: Rebbe Eliezer in the Mishneh holds that the ash of a corpse does give Tuma.