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How to include a data source that is not available in a physical file format, e.g. a web service providing real-time data streams. How to make such "data source" open access through openAIRE?

Answer: 

Depending on (i) where the data is coming from, (ii) ‘data intensity’.
If the data is OpenData already and from a reliable provider (say data.gov.uk) it is probably fine to just cite this resources. In case of doubt one might want to at least (if copyright etc. allows this) capture the stream or the historical data and deposit this as a dataset. This is definitely possible for not too large data sizes. This has the added advantage, that the analysis based on such data could be reproduced with the exact same ‘primary data’ set. In such case it might thus be useful to archive the underlying data in either a catch all or discipline specific database.
Another case might be webservice that provide an analysis on your primary data or provide computational access to a large internal dataset. As such database might go away one might want to capture exact input and output. Some providers experiment with virtual machines and docker images now, which might be a future perspective for long term preservation.