Tourismusgesetz and Beherbergungsbeiträge are just two of a plethora of terms Austria uses for accommodation taxes. The name and how much is levied varies across Austria's nine provinces.
The tax is payable on overnight accommodation (including caravans/campsites), currently ranging from €0.15 to €2.18 per person per night. For a quick overview refer to pages 24 and 25 of this pdf: Inventory of taxes.
Generally the information well presented well and easy to find. Use the links below to some comprehensive web pages and pdf documents for the relevant information on each province (be warned though, for Carinthia it is in excess of 500+ pages).
The table below lists the Austrian provinces, main destinations therein, and local terminology for the tax:-