Common cold, which is an acute viral infection of the nose, is rarely associated with fever in adults (Canadian Medical Association Journal ; DPHHS Montana).
Allergic rhinitis does not usually cause fever; it's not mentioned as a symptoms on major clinical websites (Emedicine ; Mayo Clinic ; MedlinePlus). News In Health says allergic rhinitis "never" causes fever, while The Journal Of Allergy and Clinical Immunology says that fever "is rarely found in patients with allergic rhinitis."
Non-allergic rhinitis (irritant, vasomotor, hormonal, medicamentous, gustatory, senile, etc.) is also not associated with fever according to various sources (Emedicine ; Mayo Clinic ; World Allergy Organization ; Allergy). After some further research, there seems to be no evidence that non-allergic, non-infectious rhinitis would be associated with fever, but I can't exclude the possibility that sometimes could be.
In conclusion, the absence of fever does not help (much) in determining the type of rhinitis.