Did the Greeks and Romans use reputational incentives to induce people to pay more tax?

Many public works were privately financed by completely voluntary benefactors. Those then had their names attached to buildings, like Flavian theatre, Basilica Maxentius, Baths of Caracalla etc. This is more a gift to the community in exchange for prestige. Think Carnegie Hall for modern variants of the theme.

That is very different from any form of "tax". And taxes aren't so voluntary to begin with that anyone would feel incentivised by anything.

It is also evident from history that while giving prestigious incentives for voluntary contributions might be a nice thing, but the evident systematic limitations make it quite foolish for modern people calling to reform the present system in the ancient style: to depend on philanthropy for public expenses.

From these generosities it is indeed another category to look at the actual taxes levied upon citizens in ancient times. These differed in Greece or later the Greek speaking East and Rome or later the Latin speaking East in terms os systems employed in the form of for example summae honoriae or liturgy. This ranged from 'taxes' for taking office, already a costly endavour without it, to the system most closely resembling what the question depicts: just collecting extra money, albeit with public mention of more or less enforced compliance to benefaction.

Aspects in Greece were for example the office of cultural sponsor, a Choregos. The Epidoseis:

Epidoseis (Ancient Greek: ἐπιδόσεις) was a form of non-compulsory, non-tax financial giving in ancient Greece.

These epidoseis were voluntary contributions, either in money, arms, or ships, which were made by the citizens of Athens in order to meet the extraordinary demands of the state. When the expenses of the state were greater than its revenue, it was usual for the prytaneis to summon an assembly of the people -- ecclesia -- and, after explaining the necessities of the state, to call upon the citizens to contribute according to their means.

Those who were willing to contribute then rose and said what they would give; while those who were unwilling to give anything, remained silent or retired privately from the assembly. The names of those who had promised to contribute, together with the amount of their contributions, were written on tablets, which were placed before the statues of the Eponymi, where they remained until the amount was paid.

These voluntary contributions were frequently very large. Sometimes the more wealthy citizens voluntarily undertook a "trierarchy", or the expenses of equipping a trireme. We read that the freedman Pasion furnished 1000 shields, together with five triremes, which he equipped at his own expense. Chrysippus presented a talent to the state, when Alexander the Great moved against Thebes during the Battle of Thebes; Aristophanes, son of Nicophemus gave 30,000 drachmae for an expedition against Cyprus; Charidemus and Diotimus, two commanders, made a free gift of 800 shields; and similar instances of liberality are mentioned by German classical scholar August Böckh, from whom the preceding examples have been taken.

In Latin speaking Italy things looked quite different:

Viewed as a single text our corpus of honorary inscriptions has an unmistakable dominant theme – financial generosity. Italian honorary inscriptions praise men and women who share their wealth with the public, or who spend it to the public's advantage, more than they praise any other type of meritorious individual. Building on the discussion in the last chapter, this chapter demonstrates that the type of patronage extolled and encouraged by the inscriptions' vocabulary was primarily financial. And, through close analysis of certain key virtues such as munificentia and liberalitas, it will increase our understanding of ancient Roman attitudes toward such financial patronage.

Since financial generosity receives more recognition in the inscriptions than any other type of praiseworthy behavior, one can see that money, its abundance and availability, was a basic concern to Italian municipals. This was largely because money in public treasuries was seldom used for public works and entertainment. What funds a town raised through taxes, public land rents or summae honorariae often went toward public administrative expenses, such as salaries for secretarial and menial positions, and toward the maintenance of public slaves. For public buildings, baths, games and banquets – the amenities of municipal life – communities had to turn to generous patrons. Many of these patrons financed such amenities in the form of muñera patrimoniorum, that is, duties required of members of the municipal elite to be paid for at their own expense. In some cases munera were required of all magistrates as part of their administrative responsibilities, or they were items promised by the individual while campaigning for office. Thus, the enrichment of a town's quality of life, its public image, and its morale, depended largely on the private wealth of its decurial class.

The legal discussions in Book 50 of the Digesta, however, many of which focus on exemptions from muñera or on those who do not fulfill their promises of muñera, indicate that not all members of the municipal elite were able or willing to perform public liturgies. It appears that a financial malaise had already begun to affect decurial families by the early second century. And yet, in the same period private munificence at the local level was reaching its peak, particularly early in the second half of the century. The same situation occurs in the third century, for historians of the Roman economy have underscored severe economic inflation and decline in the late second and third centuries as results of increasing demands made by corrupt imperial armies, but the evidence for Italy suggests that her municipalities continued to experience relative stability in the third century, and that certain towns even profited from privately funded building projects.

Italy was able to avoid total economic decay by virtue of her many beneficent patrons. Thus, it is all the more understandable when inscriptions honoring these individuals emphasize their munificence through precise descriptions of their expenditures and citations of their generous qualities. The majority of such inscriptions date to the late second and third centuries when the need to encourage private benefactors, particularly at the decurial level, began to increase. The smaller number of examples from the first century is partly due to a more stable economy, but also partly a reflection of conservative attitudes toward private generosity held over from the late Republic. Certainly, Cicero's contempt for large-scale public benefactions is well attested. Under the early Empire such aristocratic disdain evolved into official suspicion and disapproval. Not until Trajan's institution of the alimenta in Italy did wealthy municipals have a noble precedent for financing their own benefactions. What is most striking about all the praise of generosity in Italian inscriptions is the fact that it was aimed primarily at the municipal elite, not the imperial aristocracy; although the latter group makes up 25% of the entire corpus, they account for only about 10% of the inscriptions praising financial generosity. Italian municipalities depended mostly on the wealth of local decurial families and affluent freedmen. To be sure, these people were more intimately connected with their respective communities and, therefore, more easily influenced by verbal persuasion in honorary texts. […]

The basic concern of the average municipal ordo, as our discussion has amply demonstrated, was to supplement public tax revenues with donations from private benefactors so as to maintain certain standards of public comfort. How, then, did other dedicating groups define their own particular needs in light of the model provided by local ordines! Though their praise vocabulary is somewhat more limited in selection, it reveals the same emphasis on virtues and terms related to financial patronage. Note, for example, that the great majority (at least 80% in each case) of those inscriptions dedicated by the populace, collegia or Augustales either describe actual benefactions or make some mention of merita, munificentia, liberalitas, largitio or amor. In essence, these other dedicators present themselves as microcosms of the larger municipality and its ordo by asserting that they have the same concerns for the community's welfare.

Not only that, by using a similar vocabulary to praise generosity, as well as civic duty and morality, these groups proclaimed themselves to have the same values as those of the municipal decuriones. And in the process they portrayed themselves as being worthy of the same virtues as their social superiors. Thus, the language of praise in honorary inscriptions furnished a means by which dedicators of lower social status could bolster their public image by matching their esteem for certain virtues with that of their betters.
_{–– Elizabeth Forbis: "Municipal Virtues in the Roman Empire. The Evidence of Italian Honorary Inscriptions", Beiträge zur Altertumskunde, Vol 79, BG Teubner: Stuttgart, Leipzig, 1996.}