For the most part, no it won't. But, if you spend a lot of time in the rain, some parts could start to rust and then will either need to be cleaned or replaced.
Just make sure you wipe things dry (including chain, cassette, etc.) immediately afterwards.
I wouldn't ride in the rain if it is a heavy downpour. Water can get into places it shouldn't and it can cause some damage. I have a bike exclusively for riding in the rain. I have fenders on it and wider tires.