In the UK (and other similar nations) the answers would be no and maybe. No, you do not need any special permission when studying numismatics and philately, beyond, obviously, any restriction on your time applied by your own institution. If you wish to publish a photograph of the items there are sometimes copyright issues, or specific laws governing images of currency items. In practice they rarely apply to numismatic research and the publication would fall under fair use, but it is a matter your publisher could advise on.
It might depend upon the type of research and obviously whose (which country/government). In the United States, Title 18, Chapter 17 of US Code talks about mutilation etc of coins and currency. It also specifies that the prohibition is for fraudulently doing so. For example, destructive analysis testing to determine metal content and percentages might not fall under that provision. Photography and scanning coins should be no issue but, as Robert Bracey mentions above, there are requirements that must be followed (in the US at least) regarding size and resolution of any photography of paper currency.
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