How many mosquitoes would it take to "drain" enough blood to put a person into shock?
This question falls into different subquestions: 

1. How much blood does a mosquito take when feeding?

This is not so easy to answer, but there are publications which measure the volume of different mosquito species. Reference 1 lists volumes between 2.85 and 11.99µL per meal and mosquito. Reference 2 lists 3.07 and 5.71µL. To make the approximation a bit easier, let's assume that mosquitoes drink about 5µL per bite (which should be near enough).

2. How much blood does a human have?

According to Wikipedia, humans have around 77mL blood per kilogram of body weight. If we take a 70kg human, we get around 5L of blood.

3. How many mosquitos does it take to get a human into shock, and how many to suck him out completely?

According to the Medline, Hypovolemic shock occurs when a human loses about 1/5 (or more) of his blood volume. 1/5th of 5L is one litre. Since each mosquito sucks 5µL, this would take about 200.000 mosquitoes. To suck out all blood would take 1 million mosquitoes. (This is not possible since mosquitoes depend on an existing blood pressure, as reference 3 shows. If a capillary is empty due to massive blood sucking, they change to another.)

Besides this theoretical (but interesting) calculation, there are some real problems about getting biten by so many mosquitos. First, this is a question of available skin. The human skin is about 2 square meters in size, which would require 100.000 bites per square meter. To break this down further, a square meter has 10.000 square centimeters, so there would be 10 mosquitos which need to bite you per square centimeter. Rather unlikely.

Then each mosquito injects a mix of anti-coagulants, pain killers etc. prior to sucking to keep the blood from clotting and also to make sure you won't notice the bite immediately. As pointed out in the comments, this would already cause a lot of trouble and probably cause massive bleeding, since the effect of the anti-coagulants would sum up when biten by thousands of mosquitos. Additionally the body reacts by releasing histamines and since the amount would also be massive, this would cause real trouble and probably kill you.

References:


Regulation of blood meal size in the mosquito
La Crosse Virus Infection Alters Blood Feeding Behavior in Aedes
triseriatus and Aedes albopictus (Diptera: Culicidae)
Visualizing Non Infectious and Infectious Anopheles gambiae Blood
Feedings in Naive and Saliva-Immunized Mice