The vampires will waste a lot of blood.
Blood doesn't flow very well. At body temperature it's four times more viscous than water. As it cools it gets even more viscous at about 2% per °C. It gets worse.
Blood is a non-Newtonian fluid, it's made up of different kinds of stuff like plasma, and blood cells. This means at low flow rates these pieces can stick together and make it flow even slower sort of like jello forming out of cooling gelatin. In short, your blood river will clot. Eww.
Then there's the problem of exposure to air. Microorganisms will begin feasting on the blood. In short, it will rot. How fast this happens depends on how warm the blood is. Immune cells might help, but they're just one part of the now missing immune system.
The cells will suffocate and die. Blood cells get their oxygen by moving through tiny capillaries in the lungs only big enough to allow one or two cells to pass through at a time. This exposes maximum surface area to the oxygen-rich air filling the lungs. A river of blood does the opposite: only the cells on or immediately below the surface will be able to absorb oxygen from the air. The high viscosity means not much mixing of the blood will occur to bring deeper cells up for a "breath". Most of the blood will be oxygen deprived and die.
If they don't suffocate and die, they will starve and die. Blood cells need sugar and nutrients. There's some in the blood itself, but once they use that up no more will be coming in.
This is why when you donate blood it is never exposed to air. Blood comes out of your arm, into a tube, and into an evacuated plastic bag. The bag contains anti-coagulants are added to keep the blood from clotting, as well as sugar and nutrients to feed the blood. Then it goes into a freezer to slow its metabolism as much as possible.
@Frostfyre brought up a very good point in the comments, how much blood can be harvested sustainably? Each human has about 5 liters of blood, but if all of it is harvested the human dies and that's not very sustainable.
It's safe to donate a liter of whole blood about every 30 days. Flow rate of rivers is measured in cubic meters per second or m3/s. 1 liter is 0.001 m3 or 10-3 m3. 30 days is 2,592,000 seconds or 2.6x106 giving a flow rate of 3.8 x 10-10 m3/s per person.
There's 7 billion people or 7x109 people which means a total sustainable flow rate of 2.6 m3/s which is a decent sized creek. Here's about what that looks like.
The vampires may be able to increase the frequency they can draw blood by treating humans as cattle and force feeding them nutrients and medications to make them grow red blood cells faster, but it's never going to be much.
The vampires will be caught in a dilemma. Heat the river to make it flow and mix better and you increase how fast it rots. Cool it to preserve it as long as possible and it turns into a sticky mess. They could also increase the flow rate, but then they'd need to harvest even more blood.
The vampires could add some additives to the river to reduce clotting, "feed" the blood to keep it alive, and churn it to simulate circulation. This will buy them some time, but ultimately it's a losing battle against all the things in the air that would also love all those tasty nutrients.
What would work better is a sealed pipe.
As for its effect on the surrounding environment, that's hard to say. I don't think having a goopy, rotting mess will be a benefit.
Whole blood is about 50% water, most of it from the plasma. Fortunately, whole blood is only a very mild base with a pH of about 7.4, so it won't "salt the Earth" or the like.
My speculation is the high viscosity means it won't be absorbed too deeply in the soil. Plus a layer of dead and rotting material will cover the river bottom and banks further reducing absorption. It's my thinking that the river will have a minimal effect on the surrounding plant life.
However, insects, micro-organisms, and probably even some animals and birds will have a field day on this rich source of food.