Step 1: Understanding the Concept
A vector-borne disease is passed on by a living carrier, usually an insect.
Step 2: Check the pairs
Dengue is spread by the Aedes mosquito. Filariasis is spread by the Culex mosquito carrying Wuchereria larvae.
Step 3: Why the other options fail
Typhoid and amoebiasis spread through contaminated food and water. Ringworm spreads by contact and pneumonia by droplets. Malaria is vector-borne but typhoid is not.
Final Answer:
Dengue and filariasis form the correct pair.
\[ \boxed{\text{(C) }\text{Dengue and filariasis}} \]