I wonder if every time series data should contain noise or not. For example I am taking the price of a ticker, say Yahoo, every hour and noting the values. Does this data contains noise or not?
Periodically taken ticker data is very likely not a set of samples of a band-limited function, thus can contain aliasing artifacts which can be considered noise. This is possibly a much larger contribution than quantization and time jitter noise.
Absence of noise requires the existence of well defined signal being measured. Proving that such a signal really exists is a separate and likely far more difficult problem for some types of time series.
A huge yes. Perhaps stated in another way, due to the stochasticity, you can never be certain in practice that there is absolutely no noise in your data.
The timing at which you sample (almost periodic, every hour) might subject to small uncertainties (whether you do it by hand or by an automatic request). There is no infinite sampling precision.
Noting the values (do you note it by hand, actually?) is subject to perception and one can write figures noon accurately, forget a comma. Change $3.14$ as $31.4$ and you get a spurious peak on your time series.
Those are "copy or retrieval" noises.
As @hotpaw2 said, aliasing is a type of noise that occurs when converting a seemingly continuous data into a discrete one. It is worth pointing out (I have heard "I sample this day every 24 hour, so I have no noise". Actually, an irregular sampling might help in the case of a non band-limited signal.
Then you have almost the same issues with what Yahoo, considered here as a sensor, does. How long does it hold values? Sometimes free information is delivered at a slower pace than when it is paid? How does it round it? Does it smooth it over $3$ seconds? Moreover, does Yahoo always perform these actions in the same manner? In other words, is your signal seem stationary, or time invariant?
Those are system/sensor noises.
Finally, you take a price for a specific goal. It is a quantity that can be observed, that you may use to model a certain "real word" behavior. There might be events in the data that are "noises" for your purpose (slow trends, seasonality).
What is important is how those combined noises affect your modeling and processing.