TLDR: There are mixed results regarding its effectiveness. But it is generally considered an positive measure if the measure itself it not the policy. That is, the lottery in itself tends to be useless if no other efforts regarding the prevention of tax evasion are made.
The most recent report on the subject I could find is Taxation and the Shadow Economy (Awasthi & Engelschalk, 2018) from the World Bank. The relevant chapter is The Role of Tax Invoice Lotteries. And I quote:
Experience with lottery schemes is rather mixed, however.54 While the
costs of a lottery generally are well documented, the impact on tax
revenue collection and compliance is rarely measured in practice.55
This is partly because, as in the introduction of POS systems, clear
links cannot be established between the lottery and changes in
taxpayer compliance attitudes.
An invoice lottery is definitely unsuitable as an isolated measure to
counteract tax evasion in the cash economy. Offering occasional
substantial monetary rewards for tax invoices does not effectively
address cash-based tax evasion in some economic sectors, such as
certain service areas, and it does not reduce the risk of collusion
between buyers and sellers of a good or service. A lottery may be
useful, however, to supplement a broader public awareness campaign to
convey the importance of reducing underground economy activities and
the contributions consumers can make to achieve this objective.
There are mixed results because it is not easy to infer if it was the lottery that was effective, or just the increase in awareness for the problem. Pragmatically, at least some nations have seen increases in tax collection as can be read on Tax Lotteries: the Crowding-out of Tax Morale and Long-run Welfare Effects (Fabbri & Wilks, 2016).
Taking advantage of a recent tax lottery introduction in Portugal, we
administered a survey to a stratified sample of the Portuguese
population to verify the effects of the lottery introduction on tax
morale and tax enforcement propensity. Results confirm that tax
lotteries trigger both an increase in the number of people requesting
invoices among those individuals not used to do so and at the same
time crowd-out the willingness to engage in third-party tax
enforcement among those individuals that were accustomed to
voluntarily engage in this activity.
The EU is also documenting the experiments as can be seen in Improving VAT compliance – random awards for tax compliance (Fooken et al, 2014), but as far as I can say no official results have been reported (other than individually by the different nations).