Allergic Conjunctivitis: This condition is characterized by itchy, watery eyes and can be caused by allergens such as pollen, dust, or pet dander. The symptoms described are classic for an allergic reaction, making this the most likely diagnosis.

Dry Eye Syndrome: Although primarily associated with dryness, some forms of dry eye can cause a burning sensation and watery eyes as the body tries to compensate for the lack of moisture. Itchy eyes can also be present, especially if there's an associated blepharitis.
Blepharitis: Inflammation of the eyelids can cause burning, itching, and watery eyes. The condition often involves the corners of the eyes, where the eyelids meet, fitting the described symptoms.
Eczema (Atopic Dermatitis): Eczema around the eyes can cause itching, burning, and watery discharge. Given the description of skin burning at the corners of the eyes, eczema is a plausible diagnosis.

Angioedema: A severe allergic reaction that can cause rapid swelling of the skin and mucous membranes, potentially leading to airway obstruction. Although less common, the burning sensation and swelling could be indicative of angioedema, especially if there's significant swelling or systemic symptoms.
Keratoconjunctivitis Sicca (Severe Dry Eye): In severe cases, dry eye can lead to significant discomfort, including a burning sensation, and can compromise the integrity of the cornea. It's crucial not to miss this diagnosis due to the potential for long-term vision impairment.
Conjunctivitis due to a Foreign Body or Chemical Exposure: Chemical burns or a foreign body in the eye can cause severe irritation, including burning, itching, and watery discharge. These conditions require immediate attention to prevent permanent damage.

Trachoma: A rare infectious disease caused by Chlamydia trachomatis, leading to chronic keratoconjunctivitis. It's characterized by itchy, watery eyes and could involve the skin around the eyes, but it's less common in areas with good sanitation and healthcare.
Ocular Rosacea: A condition associated with rosacea that can cause eye irritation, including burning, itching, and watery eyes. While not as common as other diagnoses, it should be considered, especially if there are other symptoms of rosacea present.
Graft-Versus-Host Disease (GVHD): In individuals who have undergone bone marrow or stem cell transplants, GVHD can cause a range of symptoms, including ocular manifestations like dry, itchy, burning eyes. This diagnosis is rare but critical to identify due to its association with transplant patients.