I have got few white patches on neck and chin... want to know if application of salicylic acid on the spots help them turn brown? Will it work?

It will kill patches of skin for certain... thats what is used to get rid of warts.

Salicylic acid is used on acne, dandruff, psoriasis, seborrhea, calluses, corns, and warts/plantar warts, depending on the form (lotion, etc) and strength.

It won't turn white patches brown. You are missing melanin in those areas of your skin, and there's no treatment for that.