The only situation where running as fast as you can might not minimize the total amount of rain that falls on you is when the wind is from behind you. In that case, if the wind is strong enough, the optimal strategy might be to run only as fast as the raindrops are moving horizontally (if you can!).
Specifically, using the "box approximation" (that is, that a person is shaped approximately like a box) from Qmechanic's link, we can show that running as fast as you can is suboptimal if and only if all the following condition hold:
- The wind is blowing no more than 90° away from the direction you want to go.
- When standing still (and facing your destination), more rain hits you from behind than from above and from the side together.
- You're able to run fast enough that, when running, the rain only hits you from above and/or from the side.
If all these conditions are satisfied, the optimal strategy is to run only as fast as required to satisfy the third condition.
Of course, this analysis assumes not only that you're approximately box-shaped, but also that you don't care on which side of your body the rain falls on. In practice, depending on what you're wearing, you may well prefer rain on your back (or on your head, if you're wearing a hat) over rain on your face, which will affect the optimal strategy (and may well favor slower speeds in some cases).