The decision to get rid of Evrony's signature comes amid the fallout from the murder in a luxury hotel in Dubai of Mahmoud al-Mabhouh, of the Palestinian Hamas movement, last month.
Al-Mabhouh was found dead in his hotel room on January 20 after being drugged and suffocated, according to police.
'Civility and respect'
Six false Irish passports were allegedly used in the assassination.
Twelve British, four French, three Australian and one German passports were also used by 26 named suspects in the assassination, according to Dubai police, who insist Israeli spies carried out the killing.
But the Irish town's move has drawn criticism from Micheal Martin, the Irish foreign minister, who said envoys should be treated with "civility and respect".
Martin said he understood the "deep concerns" that people in Ireland harboured about some of Israel's policies, but said removing the signature went against the principles governing treatment of foreign diplomats.
"It is a basic principle of relations between states that we treat each other's diplomatic representatives with civility and respect, regardless of any policy differences," Martin told the BBC.
The Dubai police chief on Tuesday said the emirate, part of the United Arab Emirates, will use stricter screening measures before admitting Israeli dual citizenship holders into the country.
Israeli citizens are banned from travelling to the UAE, but dual nationals can use alternative passports to enter the country.
"Earlier, Israeli dual citizens could easily enter the country even if officials recognised them as Israelis from their accents and traits.
"But from now on they will be carefully scrutinised, regardless of what passport they hold," Dahi Khalfan Tamim, the police chief said.
Israel has sought to play down the row, saying there is no hard proof of its involvement.
Ireland and other European states whose passports were allegedly used in the murder have called in Israeli envoys to voice their concerns about the affair.