Did computers render useless the teaching of approximating the Binomial with Poisson and Normal distribution? When considering a binomial distribution with large $n$, it is (was?) usefull to use the Poisson or Normal distribution instead. One of the reason being the difficuty to compute the binomial coefficient for large $n$. Computers have since grown in performance and algorithms in efficiency.
Is it still worth teaching students about approximating $B(n;p)$ with $Po(np)$ for small values of $np$ (or small values of $n(1-p)$) and with $N(np;np(1-p))$ otherwise?
An ideal answer would include reference (personal or theoretical) to what is actually done in practice and to the applications of such approximations in undergraduate or graduate courses.
 A: They are indeed not so useful as calculational tools, but in terms of the theory they are important.  The approximation of binomial by Poisson motivates many applications of the Poisson distribution.  The approximation of binomial by normal leads to the Central Limit Theorem. 
EDIT:
A classic example of a practical application of Poisson distribution (from von Bortkiewicz in 1898) is the number of deaths in a year in the Prussian army due to accidental horse-kicks.  We may imagine that there are a large number $n$ of Prussian soldiers, each of whom, independent of the others, has a small probability $p$ of being accidentally kicked to death by a horse in a given year.  The total number of such deaths would then be binomial with parameters $n$ and $p$.  Since $n$ is large and $p$ small, the result should be well-approximated by a Poisson distribution, and indeed this is what von Bortkiewicz found.  Actually the probability will vary from one type of soldier to another, but this does not affect the result.
A: The Central Limit Theorem, which is the topic of your question, is still of very practical use. 
Most applications of $z$-tests and $t$-tests and several other hypothesis testing methods rely on the Central Limit Theorem for their validity. 
Take election polling, for example. Suppose you have a voting populace of zillions of people who choose from amongst two candidates $A$ and $B$, where the the proportion of $A$ voters to $B$ voters is $p$ to $1-p$, and where $p$ is not too close to either $0$ or $1$. A pollster (in the ideal setup) randomly chooses some number of voters, say 1000 of them. This sample can be thought of as a sample taken from the distribution $B(1000,p)$. The pollster then wants to apply hypothesis testing methods, i.e. the $z$-test or $t$-test, in order to determine confidence intervals for the mean, i.e. confidence intervals for the pollsters prediction of the election outcome. What justifies applying those tests is that $B(1000,p)$ is very, very close to a normal distribution.
The same justification applies every time a medical trial samples amongst patients who do or do not respond to a medication in a proportion $p$ to $1-p$.
It also applies to a particle physics experiment when sampling particles that do or do not undergo decay in a proportion $p$ to $1-p$.
A: In Grinstead and Snell's free book, Example 5.5 seems to show that the Poisson is indeed still relevant:

Suppose that in a certain fixed amount A of blood, the average human
  has 40 white blood cells. Let X be the random variable which gives the
  number of white blood cells in a random sample of size A from a random
  individual. We can think of X as binomially distributed with each
  white blood cell in the body representing a trial. If a given white
  blood cell turns up in the sample, then the trial corresponding to
  that blood cell was a success. Then p should be taken as the ratio of
  A to the total amount of blood in the individual, and n will be the
  number of white blood cells in the individual. Of course, in practice,
  neither of these parameters is very easy to measure accurately, but
  presumably the number 40 is easy to measure. But for the average
  human, we then have 40 = np, so we can think of X as being Poisson
  distributed, with parameter λ = 40. In this case, it is easier to
  model the situation using the Poisson distribution than the binomial
  distribution.

