Sponsorship deals normally qualify as publicity services, since in return for the sponsorship income some form of publicity is provided by the event organiser. In that case, the sponsorship income is subject to VAT.
The foreign organiser will in principle not have to charge VAT on its invoice, unless the sponsor is established in the same country as the organiser himself (at least when it concerns EU countries; non-EU organisers will have to consider the rules of their country).
For example, a Belgian event organiser will have to charge Belgian VAT to a Belgian sponsor, but will have to invoice without VAT to foreign sponsors in other EU countries (“reverse charge of VAT” to EU sponsors, and “place of taxation outside the EU” for non-EU sponsors).
In case also other services or goods are offered as part of the sponsorship deal, the VAT treatment will be determined based on the nature of the services or goods offered (example: a number of entrance tickets are given in return).
If no services or goods are offered in return for receiving sponsorship, i.e. the sponsor gives a certain amount and expects nothing in return, the sponsorship income will not be subject to VAT (out-of-scope).
Should the foreign organiser also re-invoice the letting of exhibition or fair space (and ancillary services), this will in most cases be subject to Belgian VAT which requires a Belgian VAT registration. It is recommended to carefully verify the letting agreement to determine the correct VAT treatment.