Contact Dermatitis: This is a common condition that occurs when the skin comes into contact with an irritant or allergen, leading to a red, swollen, and sometimes itchy rash. Given the location on the fingers and the absence of itchiness, an irritant form of contact dermatitis is plausible, possibly from repeated exposure to a chemical or substance at work or during hobbies.

Eczema (Atopic Dermatitis): Although the question mentions the rash is not itchy, eczema can sometimes present without significant pruritus, especially in its early stages or in certain locations like the hands. The chronic nature (3 weeks) could suggest an eczematous process.
Psoriasis: This chronic skin condition can cause red, scaly patches on various parts of the body, including the fingers. The lack of itch and the specific involvement of the index and middle fingers could fit a pattern of psoriatic plaques, especially if there's a history of psoriasis or psoriatic arthritis.
Infectious Causes (e.g., Paronychia): Bacterial or fungal infections around the nail or on the fingers could present with redness and swelling. The duration and lack of itchiness might suggest a low-grade infection.

Cellulitis: A bacterial infection of the skin and subcutaneous tissues that can present with redness, swelling, and warmth. It's crucial to consider cellulitis due to its potential for rapid progression and the need for prompt antibiotic treatment.
Erythema Multiforme: A skin condition characterized by target lesions, which can sometimes appear as red, swollen patches. It's often associated with infections or medications and can be a sign of a more serious underlying condition.
Lupus or Other Connective Tissue Diseases: Although less common, systemic lupus erythematosus (SLE) and other connective tissue diseases can cause skin rashes. These conditions are critical to identify due to their potential for serious systemic involvement.

Sarcoidosis: A condition that can cause skin lesions, among other systemic manifestations. Skin involvement in sarcoidosis is rare but can present in various forms, including plaques or nodules.
Granuloma Annulare: A benign condition characterized by bumps or lesions, often on the hands or feet. It's relatively rare and can be mistaken for other conditions, but it's usually harmless.
Dyshidrotic Eczema (Pompholyx): Characterized by small, itchy blisters on the hands or feet, but it can sometimes present without significant itch, making it a rare consideration in this case.