Heat Rash (Miliaria): Given the hot weather and the presence of a rash on the face and chest without fever, heat rash is a common and likely diagnosis. It occurs due to blocked sweat ducts and is prevalent in warm environments.

Eczema (Atopic Dermatitis): This is a common condition in children that can cause red, itchy rashes on the face and body. It could be exacerbated by heat and sweating.
Allergic Contact Dermatitis: An allergic reaction to something that has come into contact with the skin, such as a plant or an ingredient in a product, could cause a red rash.
Irritant Contact Dermatitis: Similar to allergic contact dermatitis but caused by an irritant rather than an allergen, leading to a non-specific skin reaction.

Kawasaki Disease: Although less likely without fever, Kawasaki disease is a critical diagnosis not to miss. It's an acute febrile illness that can present with a rash, among other symptoms like conjunctivitis and changes to the lips and tongue. Early diagnosis is crucial for treatment to prevent cardiac complications.
Staphylococcal Scalded Skin Syndrome (SSSS): This condition, caused by staphylococcal toxins, can lead to widespread skin lesions and is more common in young children. It typically starts with fever, irritability, and skin tenderness, followed by a rash.

Juvenile Dermatomyositis: An autoimmune condition that can cause skin rash and muscle weakness. The rash can appear on the face, chest, and other areas.
Lupus: Systemic lupus erythematosus is rare in young children but can cause a variety of skin rashes, including a butterfly-shaped rash on the face. It's often accompanied by other systemic symptoms.