Is it meaningful to compute a confidence interval when the sampling method is not a simple random sampling? The lower and upper bounds of the 95% confidence interval of the mean of a random variable with normal distribution of known variance $\sigma$ are $\bar{x}-1.96\frac{\sigma}{\sqrt{n}}$ and $\bar{x}+1.96\frac{\sigma}{\sqrt{n}}$. 
$n$ is the size of the sample and $\bar{x}$ is the mean value of the sample.
One assumption is that the sample of size is obtained through a simple random sampling but what does it happen when the sample is not obtained through a simple random sampling but for example through a systematic sampling? 
So, the alternative to simple random sampling is the systematic sampling where (quoting from wikipedia) the sampling starts by selecting an element from the list at random and then every $k$-th element in the frame is selected, where $k$, the sampling interval (sometimes known as the skip) is $k=\frac{N}{n}$ where $n$ is the sample size, and $N$ is the population size.
Is it meaningful to compute a confidence interval in this case? 
What are the errors in using the above formula when the sampling is not a simple random sampling?
 A: You're not specific as to which alternative to simple random sampling you have in mind. One could deliberately pick the $100$ largest observations in a simple random sample of $200$. Or the $100$ smallest. If one assumes a normally distributed population, then application of usual formula that you quote to the $100$ largest observations would hardly ever yield an interval that includes the population mean.
Or one could make the same of $200$ the $200$ largest in the whole population. That would be even worse: The validity of any conclusion you could draw would depend on the size of the population.
In any of these cases, you would not get a valid confidence interval from the usual formulas.
If you were to specify a particular alternative way of sampling then one might be able to work out a valid formula from a confidence interval when that kind of sampling is used. For example, suppose the observations are jointly normal and identically distributed and the correlation between any two of them is $1/2$. I expect one could work out a formula for use in that case. (Although just how such a distribution for the sample would arise I don't know.)
