How come water would not affect the results of a titration? We were told to rinse the buret with the base so that the concentration won't be messed up, but when it comes to the flask that is going to hold the acid, there is no need to rinse, because it would not affect the result if it has water in it.

Why is that? Wouldn't the acid be diluted a little bit and that throws off the accuracy?
The volume and concentration of base equals the number of moles of base. This is use to determine the number of moles of acid in the flask. Since you are interested in the number of moles of acid in the flask, why would adding water (which doesn't have any acid) change the number of moles of acid?