In a word, yes. I believe there are still clear situations where sampling is appropriate, within and without the "big data" world, but the nature of big data will certainly change our approach to sampling, and we will use more datasets that are nearly complete representations of the underlying population.
On sampling: Depending on the circumstances it will almost always be clear if sampling is an appropriate thing to do. Sampling is not an inherently beneficial activity; it is just what we do because we need to make tradeoffs on the cost of implementing data collection. We are trying to characterize populations and need to select the appropriate method for gathering and analyzing data about the population. Sampling makes sense when the marginal cost of a method of data collection or data processing is high. Trying to reach 100% of the population is not a good use of resources in that case, because you are often better off addressing things like non-response bias than making tiny improvements in the random sampling error.
How is big data different? "Big data" addresses many of the same questions we've had for ages, but what's "new" is that the data collection happens off an existing, computer-mediated process, so the marginal cost of collecting data is essentially zero. This dramatically reduces our need for sampling.
When will we still use sampling? If your "big data" population is the right population for the problem, then you will only employ sampling in a few cases: the need to run separate experimental groups, or if the sheer volume of data is too large to capture and process (many of us can handle millions of rows of data with ease nowadays, so the boundary here is getting further and further out). If it seems like I'm dismissing your question, it's probably because I've rarely encountered situations where the volume of the data was a concern in either the collection or processing stages, although I know many have
The situation that seems hard to me is when your "big data" population doesn't perfectly represent your target population, so the tradeoffs are more apples to oranges. Say you are a regional transportation planner, and Google has offered to give you access to its Android GPS navigation logs to help you. While the dataset would no doubt be interesting to use, the population would probably be systematically biased against the low-income, the public-transportation users, and the elderly. In such a situation, traditional travel diaries sent to a random household sample, although costlier and smaller in number, could still be the superior method of data collection. But, this is not simply a question of "sampling vs. big data", it's a question of which population combined with the relevant data collection and analysis methods you can apply to that population will best meet your needs.
The problem with studying the change of observables in time (such dataset is formally called "time series"), is that for each case observations in different points in time are dependent, which forces us at least to use tests that are valid for paired data (when comparing two data points), or VECM if you want to analyze all time points at once. In particular, significance test of regressions are invalid in such setup.
When you treat each time point as a separate independent group, you are likely to greatly exaggerate significance.
The question whether use statistical tests or not for census data depends on what is your hypothesis. If you want to infer for this particular group of people in this particular point of time (if the census actually concerns people), than there is no need for significance indeed. But very often we want implicitly to make an inference about some bigger population, e.g. whole Europe. In that case we need to treat our dataset as a sample. Or we want to make inference about our population in future; in this case we would use VECM or similar technique valid for time series.
Best Answer
It is standard practice, particularly in official statistics, to use "weighting to population" of a sample and then report estimates for the whole population, including sample error. This should not be confused with a census, but it is quite legitimate to refer to these estimates of population parameters (eg population total savings, total expenditure, etc).
Surveys are often designed specifically to facilitate this analytical approach; indeed, if they are not so designed, it can be difficult or impossible to do the appropriate weighting.